Method for transforming bandpass filters to facilitate their production and resulting devices

ABSTRACT

The invention concerns a method for optimizing elements of a narrow or intermediate band bandpass filter whereof the LC prototype has been determined. The invention is characterised in that it comprises steps which consist in: (i) breaking down several parallel or series capacitors of resonators with X elements, (ii) inserting pairs of transformers between the first and second separated element and the rest of the resonator, (iii) displacing the residual transformers to modify the impedance levels of the resonators, and (iv) absorbing the residual transformers by transformation. The invention also concerns the resulting filters.

[0001] The present invention relates to the field of pass band filters.

INVENTION GENERAL PURPOSE

[0002] The present invention relates to a process for transformation ofa circuit for the manufacture of high performance filters ormultiplexers under acceptable technical and economic conditions.

[0003] It is used to make pass band filters and multiplexers fromprototypes with different topologies with response types varying fromthe simplest to the most general, and only using inductances all equalto fixed values (or only a small number of different predeterminedvalues) to maximize performances and/or minimize the cost and/orpredetermined values of capacitances.

[0004] It is also used to make filters exclusively using dielectricresonators with the same characteristic impedance, or piezoelectricresonators with approximately or exactly the same ratio of capacitancesand with similar or equal inductances.

[0005] The process can be applied to hybrid filters or multiplexersusing inductances and dielectric resonators and/or piezoelectricresonators and obtaining the same advantages.

[0006] The essential purpose of the present invention is to obtain highperformance filters and multiplexers under better technical and economicconditions than are possible according to the state of prior art.

[0007] The design process proposed within the present invention appliesparticularly to ladder pass band filter classes with a general response(characterised by a transfer function that is a rational fraction) thatmay be obtained by synthesis (or by a circuit transformation aftersynthesis), in order to obtain a minimum inductance topology (zigzag).These filter classes are necessary in practice when it is required tomake high performance filters economically because they make moreefficient use of expensive filter elements. The process is equallyapplicable to other filter topologies including filter topologiesconventionally used to make polynomial filters.

[0008] The technique proposed within the framework of the presentinvention comprises three main variants applicable particularly tofilters using inductances and capacitances, filters with dielectricresonators and filters using piezoelectric resonators with bulk orsurface waves that will be described one after the other.

GENERAL CONTEXT OF THE INVENTION

[0009] Passive filters with LC circuit or dielectric resonators, orsurface or bulk wave acousto-electric resonators are widely used intelecommunications and particularly in radioelectric transmissions or oncable to limit the signal spectrum or for switching to enable sending(and extraction) of several signals occupying different frequency bandson the same antenna or on the same transmission support. In most cases,they are filters used in large numbers for which high performances arerequired at the lowest possible cost.

[0010] The cost of filters, and often their insertion losses, dependslargely on the number of inductances (and/or resonators) used sinceinductances and the various resonator types are significantly moreexpensive than capacitors. For a given number of these elements, thecost is very strongly reduced if the filters only comprise one standardmodel. It is further reduced if the only distinction in the manufactureof the expensive elements used is in the use of an inexpensive andnecessarily individual technological step (for example this is the casefor the frequency adjustment of dielectric resonators with the samecharacteristic impedance, or piezoelectric resonators with the samedesign).

[0011] Note also that dissipation from inductances and TEM dielectricresonators is usually greater than from capacitances, and that there areranges of values or of parameters that minimize their losses for eachfrequency.

[0012] Thus, for a given technique, the range in which inductances orresonators with good characteristics (overvoltage, parasite elements,etc.) can be made is much narrower than the range in which capacitanceswith favourable characteristics can be made. These elements are alsomade in smaller quantities, and in prior art often had to be madespecially to make a given filter.

[0013] More precisely, there is only a fairly narrow range of inductancevalues that can be bought off-the-shelf or specially made for which goodperformances (overvoltage, natural resonant frequency, etc.) can beobtained at an attractive cost, for frequencies higher than a fewMegahertz.

[0014] Filters calculated by conventional methods usually includeinductances with different values, and therefore are unlikely tocorrespond to standardized values available off-the-shelf.

[0015] In practice, for TEM dielectric resonators, there are only a fewvalues of permittivity and a small number of dimensions (sections) andtherefore characteristic impedances of resonators.

[0016] For resonators that use a bulk or surface acousto-electricinteraction, it is only known how to make devices with a small number ofvalues of the ratio of the two capacitances (as a first approximation,this magnitude only depends on the material (for example quartz), thewave type and the crystalline orientation), and for a given frequency,inductances located within a very narrow range of values. In practice atthe present time, this state of affairs reduces the number of circuittopologies used to make crystal filters.

[0017] Therefore, it is very useful to find techniques that only use afew (if possible only one which will optimise the cost) well chosenvalue(s) for all inductances, in inductance filters and capacitances.Similarly, for dielectric resonator filters, it is necessary to use onlythe few available values of characteristic impedances (only one willgive the most economic solution, if possible). Similarly, for apiezoelectric resonators filter, it must be possible to use devices witha value very close to a given value of the capacitances ratio and aninductance located within a narrow range. It is also very useful to makeladder crystal filters with a general response, since sensitivityproperties to dispersions obtained for this topology are better thandispersions obtained for differential structure filters.

[0018] For obvious reasons of efficiency, the invention broadly coversfilter classes that make very good use of inductances (or various typesof resonators); in this presentation, efficiency is understood as beingcharacterised by selectivity, for example measured by a shape factorobtained for a given number of inductances or resonators.

[0019] The most attractive of these filter classes are called “minimuminductances” filter classes, and have a ladder topology. This “zigzag”topology (using an extended meaning beyond the definition of reference[5]), is called zigzag because (in the discrete elementsrepresentation), it is composed of an alternation of horizontal andvertical arms composed of irreducible three-element dipoles (oneinductance and two capacitances forming a resonator with a seriesinductance (Z=0) and a parallel resonance (Y=0)), possibly except forthe arms at the ends. These resonators themselves can also be arrangedaccording to two topologies (known transformation formulas can be usedto change from one to the other, and will be described later). Thedipoles at the ends of the filter may comprise one, two or three L, Celements. The zigzag filters are considered as being technologicallybetter [1] [2] [3] than the more usual ladder topology filters (exceptfor a few reserves for some cases, possibly except for filters with thissame ladder shape and that would comprise more complex irreducibledipoles than those with three elements in each of their arms). Theselatter filters have not been studied very much, and rarely have anyadditional interest. Furthermore, unlike the case of dipoles with threeelements, there is only a small number of distributed dipoles that canbe represented by such equivalent circuits and that have a particularinterest (in practice, only multimode distributor resonators can haveany particular interest).

[0020] Filter classes [1] [5] with a zigzag topology thus defined(Skwirzynski classification based on the properties of the numerator anddenominator polynomials of the transfer function S(p) and thecharacteristic function Φ(p), and more particularly the functions thataffect the behaviour at zero and infinite frequencies), includeparticularly the even order elliptical filters (equal ripple in passband and in attenuated band) that make the most efficient use ofinductances to approximate a rectangular envelope curve (a Chebyshevapproximation both in pass band and in attenuated band). For thesefilters, the zigzag topology is obtained by synthesis or from more usualtopologies using the Saal and Ulbrich [2] transformation. It includes LCresonators with two elements of two different types at the two ends.

[0021] Other filter types with the same topology or with a similartopology (that are different due to the dipoles at the ends) are alsoknown. The responses of some filters are more attractive than theresponses of the above filters in cases in which it is desirable to haveasymmetry of the slopes of the transition bands, or differentattenuations in the two attenuated bands, and different behaviours atf=0 and at f=∞.

[0022] Ladder filters composed entirely of three-element dipoles (zigzagin the strict sense for reference [5]) also have the advantage of havingthe maximum number of attenuation poles at finite frequencies that canbe made simply with a given number of inductances, the properties offunctions forming their distribution matrix are such that they have someminor theoretical and practical disadvantages compared with the previousfilters (including more complex calculation and manufacture). However,like all topologies composed only of this type of resonator andcapacitance, their main usefulness is for making filters usingdistributed elements that can be represented by this type of circuits,at least close to some of their characteristic frequencies.

[0023]FIG. 1 shows an example of a minimum inductance filter (zigzag).

[0024] This FIG. 1 shows two end quadripoles A and B separated by fourdipoles R₃₁, R₃₂, R₃₃, R₃₄.

[0025] In this case, quadripole A comprises an inductance and acapacitance placed in parallel between two horizontal conducting arms(two-element dipole).

[0026] In this case, quadripole B comprises an inductance and acapacitance in series on the horizontal upper arm, and a horizontallower conducting arm (also a two-element dipole).

[0027] Dipoles R₃₁ and R₃₃ are placed in series on the upper horizontalarm between the two quadripoles A and B.

[0028] Dipole R₃₃ forms a vertical arm placed between the two dipolesR₃₁ and R₃₃. The dipole R₃₄ forms another vertical arm placed betweendipole R₃₃ and quadripole B.

[0029] Each of the four quadripoles R₃₁, R₃₂, R₃₃ and R₃₄ comprises aninductance and two capacitances (three-element dipole) arranged:

[0030] either in the form of a capacitance in series with a circuitcomprising a second capacitance in parallel with an inductance,

[0031] or in the form of a capacitance in parallel with a circuitcomprising a second capacitance in series with an inductance,

[0032] as shown diagrammatically in FIG. 2.

[0033] The circuits for the zigzag filters obtained directly bydifferent synthesis methods (or by a transformation) almost alwayscontain inductances (or equivalent inductances for dielectric orpiezoelectric resonators) with very different values.

[0034] Pass band multiplexers are sets of filters with at least one (andsometimes n) common access and p other accesses and performing selectiveswitching of signals contained in a given frequency band from an access(for example the single common access) to another. Known methods ofcalculating multiplexers are usually based on digital optimisationtechniques.

STATE OF PRIOR ART

[0035] Minimum inductance filters have been known for a long time (forexample see the book by Zverev [3]). Elliptical filters (Cauer) havealso been known for a long time, and have been described in manyarticles and tabulated by many authors (for example see [2] and [3]).The method proposed by P. Amstutz [4] and the method of transformingconventional topologies of even filters into minimum inductance filters(zigzag) as proposed by Saal and Ulbrich [2], are often used for thesynthesis of these filters.

[0036] The design of filters with “infinite” attenuation points atfrequencies fixed in advance and with equal ripple in the pass band bytransformation in z is described in reference [2] that is used todetermine the characteristic polynomials of the filter and to synthesizethem using methods clearly described in the same article and describedin the book by Mr. Hassler and Mr. Neyrinck [5] and also in theTechniques de l'Ingénieur (Engineering Techniques) [6] collection.

[0037] It is also important to remember that there are some knownexamples of methods of synthesizing polynomial pass band filters withvery narrow bands, so that they can use inductances with the same valuesor identical resonators (P. Amstutz [7]). These methods usually useimpedance inverters (or admittance inverters) that can only be madeapproximately by quadripoles with known discrete passive elements (validapproximation within a very narrow frequency band). Note also that whenan attempt is made to economically obtain high performances, polynomialfilters are less attractive since their performance is not as good forthe same number of inductances (or resonators) as filters with moregeneral responses.

[0038] Impedance or admittance inverters can be made fairlyapproximately by the use of distributed elements, and the use of suchcomponents is fairly natural in filters using resonators that arethemselves distributed. For example, the design of filters withimpedance inverters with determination of the transfer function to havethe same ripple in the pass band with “infinite” attenuation points atfrequencies given in advance and a direct synthesis, is described by R.J. Cameron in references [8], [9], whereas a precise example of makingimpedance inverters was given by G. Macchiarella [10].

[0039] Note also that there are many other techniques for calculatingdielectric resonator and piezoelectric resonator filters. Thiscalculation is usually made starting from the calculation of inductancefilters and capacitances by circuit transformations leading to theappearance of equivalent circuits with discrete elements of thesecomponents making use of different approximation techniques. References[11 to 15] describe elements concerning dielectric resonators andparticularly resonators using transverse electromagnetic (T.E.M.) modesof dielectric solids with simple geometries (usually coaxial) and theyare used to make filters used in the field of mobile radiocommunications. References [16 to 20] describe principles governing theoperation of piezoelectric resonators with bulk and surface waves, andprinciples about their use to make pass band filters.

[0040] Furthermore, there are several known techniques for thetransformation of circuits (Norton, Haas, Colin transformations, etc.).However, apart from Saal and Ulbrich transformation that only aims at atopology modification, transformations of LC filter circuits are alwayspresented as being made on only some elements of the filter, and notsystematically [23 to 26].

[0041] The calculation of pass band multiplexers has almost always beenbased on digital optimisation techniques, and has been known for sometime [27]. However, a great deal of recent progress has been made withthis technique [28-29]. Apparently there has been no attempt to makesystematic circuit transformations on these devices.

[0042] Despite the very abundant literature available on filters, nofully satisfactory technique has yet been proposed for their design.

[0043] Due to the lack of a systematic approach in the past, circuittransformations were only made to modify the values of some componentsof a filter (often the inductances). Known transformations were usuallymade manually, and additional degrees of freedom that could have beenintroduced by separating components were not used systematically.

[0044] One particular result was that the real advantages of “minimuminductance” filters were rarely used for LC filters with inductances andcapacitances for frequencies higher than about 10 MHz, and even morerarely (no known examples to the inventors) for dielectric resonatorfilters or piezoelectric resonator filters. This was mainly due to thefact that filters made using partial transformations, made only tomodify a few values of inductances (in practice impossible to achieve orthat excessively degraded performances), were not sufficientlyattractive economically, or did not make a sufficient improvement to theease of manufacturing.

[0045] In particular, work done by the inventors shows that the only wayof obtaining an important economic advantage while facilitatingmanufacturing is by a global consideration of all transformations andthe simultaneous use of additional degrees of freedom possible usingthis global approach.

DESCRIPTION OF THE INVENTION

[0046] The systematic and global process for transformation of “minimuminductance” pass band filters and several other filter topologiesaccording to the present invention, is intended to obtain filters usingonly one (or few) determined value(s) of inductances, or usingdielectric resonators with the same characteristic impedance, or usingpiezoelectric resonators with approximately or exactly the samecapacitance ratios and approximately or exactly the same inductances.This process also has advantages other than those mentioned above, suchas the choice of equal values for some capacitances or the choice oftypical off-the-shelf values for some capacitances. The process isequally applicable under similar conditions to connection filterscomposed of filters made according to previously mentioned technologies.

[0047] The process will be described principally for the case of filterswith a minimum number of inductances that are among the most generalfilters, and in practice among the most useful for economicallyobtaining high performances. We will mention later on that, without anysignificant change to the principle the process is applicable to othersimpler filter topologies (for example polynomial pass band filtertopologies) that, although they do not use inductances or resonators asefficiently as the previous filters, may be attractive in some specialcases.

[0048] In order to get a clear understanding of the invention, rememberthat three-element resonators (one inductance and two capacitances) mayoccur in one of the two forms illustrated in FIG. 2 comprising acapacitance C_(sa) in series with a parallel resonator LC formed fromthe capacitance C_(pa) and the inductance M_(pa), or in a dual manner byputting a capacitance C_(pb) in parallel and a series resonator formedby the inductance L_(sb), in series with the capacitance C_(sb). Byconvention, the first form a will be called a “series form” and thesecond form b will be called a “parallel form” in the rest of this text.

[0049] The following relations show the correspondences between the twoforms “a” and “b”:

C _(pa) /C _(sa) =C _(pb) /C _(sb)=“ratio of resonator capacitances”with three elements

M _(pa) C _(pa)ω_(a) ²=1 L _(sb) C _(sb) ω _(b) ²=1

ω_(b) ²=ω_(a) ²/[1+C _(sa) /C _(pa)]=ω_(a) ²/[1+C _(sb) /C _(pb)]

C _(sa) =C _(pb) +C _(sb) ; C _(pa) =C _(pb)[1+C _(pb) /C _(sb) ]; M_(pa) =L _(sb)[1+C _(pb) /C _(sb)]²

C _(pb) =C _(pa)/[1+C _(pa) /C _(sa) ]=C _(sb) +C _(sa)/[1+C _(pa) /C_(sa) ] L _(sb) =M _(pa)[1+C _(pa) /C _(sa)]²

[0050] where ω_(a) and ω_(b) are the anti-resonant frequency (Y=0) andthe resonant frequency (Z=0) angular frequencies of the losslessresonator. These dipoles are characterised by three parameters that canbe chosen in different manners. The capacitances ratio is a usefulparameter because it is independent of the “impedance level” (forexample characterised by the value of the inductance) and it dependsonly on the ratio of remarkable frequencies and their relativedifference.

[0051] We will describe the process according to this invention firstlyby considering the example of the case of a zigzag filter with atopology similar to the topology indicated in FIG. 1, for which thenumber of three-element resonators is K (even) and that has tworesonators with two elements at the ends as shown in FIG. 1. Theproposed process is also applicable to cases of zigzag filters withother types of dipoles at the ends (comprising not more than oneinductance) and also to the case of usual topologies of polynomialfilters (sequence of series LC dipoles in horizontal arms and parallelLC dipoles in vertical arms). It is also more generally applicable tothe case of ladder filters comprising a maximum of one inductance perarm and satisfying conditions that will be specified later.

[0052] The process according to this invention for systematicmodification of impedance levels of resonators in a synthesis LC filteror obtained by transformation from a prototype as described above ispreferably based on the following steps:

[0053] A/Decomposition of Capacitances:

[0054] Decomposition into three parts of parallel capacitances C_(p) orseries capacitances C_(s), into resonant circuits with two LC elementsin parallel or in series (in this case located at the ends of thefilter), for example as illustrated in FIGS. 3a and 3 b.

[0055] The decomposition illustrated in FIG. 3a consists of replacing aparallel circuit comprising an inductance L_(p) in parallel with acapacitance C_(p) by a circuit comprising an inductance L_(p) inparallel with three capacitances C_(pu)/C_(pv) and C_(pw).

[0056] The decomposition illustrated in FIG. 3b consists of replacing aseries circuit comprising an inductance L_(s) in series with acapacitance C^(s) by a circuit comprising an inductance L_(s) in serieswith three capacitances C_(su), C_(sv) and C_(sw).

[0057] The result is:

C _(pa) =C _(pa u) +C _(pa v) +C _(pa w) (input side A)

C _(sb) =C _(sb u) ⁻¹ +C _(sb v) ⁻¹ +C _(sb w) ⁻¹ (output side B)

[0058] Decomposition of series capacitances C_(s) associated with thethree-element resonators (horizontal) in series form, into three parts,for resonator number q (as illustrated in FIG. 4a). This gives C_(s q)⁻¹=C_(s qu) ⁻¹+C_(s qv) ⁻¹+C_(s qw) ⁻¹.

[0059] The decomposition illustrated in FIG. 4a consists of replacing acircuit comprising a capacitance C_(s) in series with a resonator formedfrom an inductance M_(p) in parallel with a capacitance C_(p), by acircuit comprising two capacitances C_(su), C_(sw), in series on theinput side of a resonator formed from an inductance M_(p) in parallelwith a capacitance C_(p) and a third capacitance C_(sw) in series on theoutput side of this resonator.

[0060] Decomposition of shunt capacitances C_(p) associated with thethree-element resonators (vertical) in parallel form into three parts,for resonator number r (as illustrated in FIG. 4b). This givesC_(pr)=C_(pru)+C_(prv)+C_(prw).

[0061] B/Insert pairs of transformers with ratios 1/m_(i) and m_(i)/1between the first element thus separated with subscript u and the restof the resonator, for example between C_(p ru) and the residual shuntresonator formed from L_(sr) in series with C_(sr), with C_(p rv) andC_(p rw) in parallel (see FIG. 5). One pair of transformers will beinput for each inductance to be transformed.

[0062] C/Displace a transformer from each pair to the next resonatortransformer (FIG. 6). The consequences are to:

[0063] modify impedance levels of resonators (which is one essentialpurpose of the invention),

[0064] make transformers with ratios such as m₁/m₂=1, then m₂/m₃=1 thenm₃/m₄=1, . . . , etc. appear, and to modify the value of the filterclosing impedance at end B.

[0065] D/Eliminate internal transformers by Norton equivalence, andreplace these transformers and two capacitances determined in theprevious decomposition and the values of which were modified bydisplacement of transformers, by two other capacitances. For filterswith the topology given in FIG. 2, the transformation relations give2*(N/2−1) relations (N=filter order, which in this case includes a totalof N/2 resonators) relating the values of the new capacitances to theold capacitances and to transformation ratios and conditions ontransformation ratios (m_(i)/m_(i+1)) such that these transformationsare possible (positive signs of transformed elements).

[0066]FIGS. 7a and 7 b diagrammatically show two of thesetransformations, according to the original configuration ofcapacitances.

[0067] The transformation illustrated in FIG. 7a consists of replacing acircuit comprising a parallel capacitance C_(p) on the input side and aparallel transformer on the output side, connected through a seriescapacitance C_(s) on the head arm by a circuit comprising a seriescapacitance C_(st) on the input side and a parallel capacitance C_(pt)on the output side.

[0068] For the case in FIG. 7a, if n is the transformation ratio(n=m_(i)/m_(i+1), where i is odd):

[0069] The imposed condition necessary to be able to make thetransformation is:

C _(p) =C _(piw) /m _(i) ² =C _(s)[(m _(i) /m _(i+1))−1]=(C _(si+1 u) /m² _(i))[(m _(i) /m _(i+1))−1]

[0070] where C_(piw)=C_(si+1u)(m_(i)−m_(i+1))/m_(i+1)

[0071] and m_(i)>m_(i+1) or n>1.

[0072] The values of elements resulting from the transformation arethen:

C _(st) =C _(si,i+1) =C _(piw) /[m _(i)(m _(i) −m _(i+1))]=C _(si+1u)/(m_(i) −m _(i+1))

C _(pt) =C _(pi,i+1) =C _(piw) /m _(i)(m _(i) −m _(i+1))=C_(si+1u)(m_(i) −m _(i+1))/(m _(i+1) ² m _(i))

[0073] The condition necessary to only have positive elements is thenthat n>1 or m_(i)>m_(i+1) (where i is odd and for a zigzag filteraccording to FIG. 1).

[0074] The transformation illustrated in FIG. 7b consists of replacing acircuit comprising a series capacitance C′_(s), on the input side and aparallel capacitance C′_(p) on the output side with a paralleltransformer on the output side, by a circuit comprising a parallelcapacitance on the input side C′_(pt) and a series capacitance on theoutput side C′_(st).

[0075] For the case shown in FIG. 7b, if n′ is the transformation ratio(n′=m_(i)/m_(i+1) where i is even):

[0076] The imposed condition necessary to be able to make thetransformation is:

C′ _(s) =C _(siw) /m _(i) ² =C′ _(p) [m _(i)/(m _(i+1) −m _(i))]=(C_(pi+1u) /m ² i)[m _(i) /m _(i+1) −m _(i))]

[0077] The values of elements resulting from the transformation arethen:

C′ _(si) =C _(si,i+1) C _(p) _(i+1u) [m ² _(i+1)/((m _(i+1) −m _(i)).m_(i))]

C′ _(pt) =C _(pi,i+1) =C _(p(i+1)u)/(m _(i) .m _(i+1))

[0078] The necessary condition to obtain only positive elements is thatn<1, namely m_(i)<m_(i+1) (if i is even and for a zigzag filteraccording to FIG. 1).

[0079] E/Elimination of Transformers From Ends

[0080] By transformation using known narrow band equivalences (forexample see references [7] and [23 to 26]), it is possible to replacetransformers remaining at the ends of the filter by two impedances withopposite signs (C and L or C and −C′ or L and −L′) forming a Gamma (Γ)or a Tau (τ) depending on the transformation ratio. Steps are frequentlytaken such that one of the components located at the corresponding end(or a capacitance resulting from the decomposition of the originalcapacitances into three parts, or an original inductance for which thevalue was modified by displacement of transformers) is absorbed in thistransformation or is replaced by another component with the same natureand with a different value. The result is that the number of componentsis not significantly increased (which is a means of absorbing negativevalues that can occur for example in the separation of capacitances).

[0081] In all cases, the transformation relations are written (there aretwo transformation relations at each end) together with the relationrelated to absorption or recombination of a component, if there is sucha relation.

[0082] If Z₀ is the required termination impedance, and if Z_(s) andZ_(i) are the termination impedances of the zigzag prototype circuitafter displacement of transformers at the ends, the terminationimpedances are Z*=Z_(s)/m₁ ² and Z*=Z_(i)m² _(k) respectively, if thereare k inductances in the prototype. Either the transformation in FIG. 8aor the transformation in FIG. 8b (derived from each other) may be used,depending on whether Z* is greater than or less than Z₀.

[0083] In both cases, matching is done using an impedance Z₁ in serieson a horizontal branch, and an impedance Z₂ in parallel on a verticalbranch.

[0084] For the case in FIG. 8a, matching is done using impedances Z₁ andZ₂ with opposite signs:$Z_{1} = {{{\pm j}\sqrt{Z_{0}\left( {Z^{*} - Z_{0}} \right)}\quad {and}\quad Z_{2}} = {{\pm {jZ}^{*}}{\sqrt{\frac{Z_{0}}{Z^{*} - Z_{0}}}.}}}$

[0085] For example, for end A of the zigzag filter in FIG. 1, we haveZ*=Z_(s)/m₁ ², if Z_(s)m₁ ²>Z₀ and if the impedance Z₁ is chosen to be acapacitance, then Z₂ will be a negative capacitance that will besubtracted from C_(p 1u) and C_(p 1v) or an inductance in parallel withthe transformed inductance L_(a). This transformation is made for thecentral frequency of the filter (ω=ω_(c)).

[0086] Then:

Z ₁ =−j/C _(sa) ω=−j{Z ₀(Z _(s) m ₁ ² −Z ₀)}^(1/2) or C _(sa)=1/{ω² Z₀(Z _(s) m ₁ ² −Z ₀)}^(1/2)

Z ₂ =+jZ _(s) m ² ₁ *{Z ₀/(Z _(s) m ₁ ² −Z ₀)}^(1/2) or C _(pa)=−{(Z_(s) m ₁ ² −Z ₀)/Z ₀}^(1/2)/(ω.Z _(s) .m ₁ ²)

[0087] The case in FIG. 8b can easily be transposed from the previouscase: $\begin{matrix}{Z_{1} = {{\pm j}\sqrt{Z_{0}\left( {Z^{*} - Z_{0}} \right)}}} & {{{and}\quad Z_{2}} = {{\pm j}\quad Z_{0}\sqrt{\frac{Z^{*}}{Z_{0} - Z^{*}}}}}\end{matrix}.$

[0088] For example, for end B of the zigzag filter in FIG. 1, the resultis Z*=Z₁.m_(k) ² if Z₀>Z*, and if it is chosen that the impedance Z₂should be a capacitance, Z₁ will be an inductance that will be added tothe transformed inductance L_(ab) (L_(sb)), or a negative capacitancethat can be recombined with the capacitance C_(bw)+C_(s a w).$\begin{matrix}{Z_{1} = {{+ j}\sqrt{Z^{*}\left( {Z_{0} - Z^{*}} \right)}}} & {{{or}\quad L_{sba}} = {\frac{1}{\omega}\sqrt{Z^{*}\left( {Z_{0} - Z^{*}} \right)}}} \\{Z_{2} = {{- {jZ}_{0}}\sqrt{\frac{Z^{*}}{Z_{0} - Z^{*}}}}} & {{{or}\quad C_{pb}} = {\frac{1}{Z_{0.\omega}}\sqrt{\frac{\left( {Z_{0} - Z^{*}} \right)}{Z^{*}}}}}\end{matrix}$

[0089] Naturally, m² _(k) can also be determined such thatL_(sb)+L_(sba) is equal to a predetermined value (equation in m² _(k)which will for example replace the equation L_(sb)=L_(b) m² _(k)).

[0090] In some cases, for example concerning relatively wide filters, itmay be useful to replace the previous technique using a “wide band”matching method that uses more components but that provides anon-negligible filter complement (particularly if pass band networks areused for this purpose). These known techniques described in differentbooks may also take account of partially reactive terminations (forexample in UHF, resulting from accidental distributed capacitances).

[0091] F/The equations thus obtained show the number of degrees offreedom in fixing the values of some elements of the filter.

[0092] Operations D are used to express an essential relation betweenthe values of the capacitances of the decomposition of a pair ofconsecutive resonators, for example between C_(p 1 w) and C_(s 2 u).This is applicable for a model filter with N (order 2N) resonatorsgiving a number N+1 of relations (2 for impedance levels at the ends andN−1 for relations between the N adjacent resonators). We also have therelations of the decomposition of capacitors described in A) above intothree, giving N relations for N resonators. These 2 N+1 relations do notcompletely determine the 3 N capacitances with subscripts {u, v, w} andthe N transformation ratios. Therefore, there are still 2 N−1 degrees offreedom to fix the values of filter manufacturing elements.

[0093] Much of the advantage of the process proposed within theframework of this invention is the potential use of these additionaldegrees of freedom to make it more economic to make LC filters, or toenable the use of distributed dipoles that can be manufacturedeconomically and/or which have more attractive characteristics within agiven frequency range, under favourable conditions.

[0094] Thus, this invention proposes a process for optimisation of thevalues of the elements of a narrow or intermediate pass band filter forwhich the LC prototype has been determined, characterised by the factthat it includes the following steps:

[0095] i) decompose parallel or series capacitances of two- orthree-element resonators, into two or three elements,

[0096] ii) insert pairs of transformers between the first separateelement and the rest of the resonator,

[0097] iii) displace transformers to modify resonator impedance levels,and

[0098] iv) absorb residual transformers by transformation.

BRIEF DESCRIPTION OF THE FIGURES

[0099] Other characteristics, purposes and advantages of this inventionwill become evident after reading the detailed description given below,with reference to the attached drawings, given as non-limitativeexamples in which:

[0100]FIG. 1 shows an example of a conventional minimum inductance(zigzag) filter,

[0101]FIG. 2 shows two topologies of three-element dipoles belonging tothis filter,

[0102]FIGS. 3a and 3 b show the decomposition of capacitances of dipoleswith two LC elements for a parallel configuration and for a seriesconfiguration respectively,

[0103]FIGS. 4a and 4 b show the decomposition of three-element dipolecapacitances in the series form and in the parallel form respectively,

[0104]FIG. 5 diagrammatically shows the transformer insertion step,

[0105]FIG. 6 diagrammatically shows the transformer displacement step,

[0106]FIG. 7a and FIG. 7b diagrammatically show the step to eliminateinternal transformers, for two capacitance configurations,

[0107]FIG. 8 diagrammatically shows the step to eliminate transformersat the ends of the filter, for Z*>Z₀ in FIG. 8a and for Z*<Z₀ in FIG.8b,

[0108]FIG. 9 shows the circuit for a prototype zigzag filter obtained bysynthesis,

[0109]FIG. 10 shows the electrical circuit corresponding to thetransformed filter,

[0110]FIG. 11 shows the responses of the prototype filter compared withthe responses of the transformed filter,

[0111]FIG. 12 shows the circuit of a filter to which the processaccording to this invention is applicable (note that it includes“sub”-circuits with different topologies),

[0112]FIG. 13 shows the circuit for an order 12 prototype filter,

[0113]FIG. 14 shows the circuit for a transformed filter with ATNTNBtopology (note, the topology is marked starting from end A of the filterand working towards end B, using the letters N and T to indicate thatthe shape of the three-element dipoles was conserved (N) or transformed(T)),

[0114]FIG. 15 shows the circuit for a transformed filter with ANTTNBtopology,

[0115]FIG. 16 shows the circuit for a transformed filter with ANTNTBtopology,

[0116]FIG. 17 shows the circuit for a transformed filter with ATNTTBtopology,

[0117]FIG. 18a shows the response of the prototype filter and theabove-mentioned lossless transformed filters,

[0118]FIG. 18b shows details of the pass band response of the prototypefilter and the above-mentioned transformed filters (it will be observedthat the curves are exactly superposed),

[0119]FIG. 19 shows a prototype circuit used for a transformation,

[0120]FIG. 20 shows the structure of a first resonator for this circuit,

[0121]FIG. 21 shows the transformation structure for a second resonator,

[0122]FIG. 22 shows the structure of a third resonator,

[0123]FIG. 23 shows the structure of a final resonator,

[0124]FIG. 24 shows an example of a complete solution aftertransformation,

[0125]FIG. 25 shows the response of the transformed filter for lines ofequal characteristic impedance,

[0126]FIG. 26 shows the circuit for an order 12 prototype zigzag filter,

[0127]FIG. 27 shows the response of such a prototype zigzag filter,

[0128]FIG. 28 shows the circuit for a transformed filter with matchingby two impedances,

[0129]FIG. 29 shows the response of the transformed filter for the TEMresonators, Zc=11.5 Ohm (matching for inductances at both sides),

[0130]FIGS. 30a and 30 b show the responses of the transformed filterrespectively with matching by capacitances, in pass band in FIG. 30a andfor the entire response in FIG. 30b,

[0131]FIG. 31 shows a circuit equivalent to a resonator with lithiumtantalate,

[0132]FIG. 32 shows the circuit for an initial zigzag filter,

[0133]FIG. 33 shows the circuit for the transformed filter,

[0134]FIG. 34 shows the response of a crystal filter obtained bytransformation (lossless),

[0135]FIG. 35a shows the response of a transformed filter taking accountof losses, and FIG. 35b shows the response of the transformed filtertaking account of losses in pass bands,

[0136]FIG. 36 shows the circuit for the transformed filter,

[0137]FIG. 37 shows the response of an order 8 transformed filter withtwo L-SAW resonators and inductances (taking account and not takingaccount of losses), the response without losses is coincident with theresponse of the prototype)),

[0138]FIGS. 38a, 38 b and 38 c show three frequently used topologies forresonator filters.

[0139] The following figures are applicable for manufacturing of theduplexer according to the invention (connection filter with two inputsto one output).

[0140]FIG. 39 shows the response of two prototypes taken in isolation,

[0141]FIG. 40 shows the variation of responses in pass band duringoptimisation (the two filters then have a common end),

[0142]FIG. 41 shows responses finally obtained for the duplexer,

[0143]FIG. 42 shows the impedance transformation at the end common tothe two filters,

[0144]FIG. 43 shows the transformed circuit of a duplexer,

[0145]FIG. 44 shows the electrical response of a duplexer aftertransformation of the circuit shown in FIG. 43,

[0146]FIG. 45 shows the transformed circuit of a duplexer,

[0147]FIG. 46 shows a circuit comprising connection filters that can bemade according to this invention (for a mobile radiotelephonyapplication for equipment that will have to operate in multiplefrequency bands),

[0148]FIG. 47 shows another example embodiment conform with thisinvention showing which functions can be made in a mobile terminal or amulti-frequency and multistandard base station,

[0149]FIG. 48 diagrammatically shows the transformation of T or Πsub-circuits into Π or T sub-circuits respectively, and

[0150]FIGS. 49 and 50 show two transformation variants conform with thisinvention.

DETAILED DESCRIPTION OF THE INVENTION

[0151] 1. Application to LC Filters with Inductances with Values Fixedin Advance

[0152] In this case, the transformation principle is as follows.

[0153] To illustrate the principle described above, we will start withan order 8 filter with a pass band between 890 and 905 MHz and infiniteattenuation points fixed at 885 and 910 MHz, with a behaviour whichwould give a pass band insertion loss ripple equal to 0.3 dB for idealinfinite overvoltage elements.

[0154] The corresponding zigzag filter circuit as obtained by a knownsynthesis method is shown in FIG. 9.

[0155] The value of the elements in the prototype circuit is given inTable 1 below (MKSA units, capacitances in Farads, inductances inHenry).

[0156] For three-element resonators, the subscripts a and b in thenotations correspond to the forms a and b respectively as defined inFIG. 2. TABLE 1 ##zigzag filter G08 RS = 106.54 1 RL = 75.00 6 1 CAP Cpa= Ca 1 0 0.11436E−09 2 IND Lpa = La 1 0 0.27327E−09 3 CAP Csa2 = C1 1 20.89352E−12 4 IND Mpa2 = L2 2 3 0.77700E−09 5 CAP Cpa2 = C2 2 30.39361E−10 6 CAP Csb3 = C3 3 4 0.99994E−13 7 IND Lsb3 = L3 4 00.32340E−06 8 CAP Cpb3 = C4 3 0 0.44050E−11 9 CAP Csb = Cb 3 50.34418E−13 10 IND Lsb = Lb 5 6 0.91955E−06

[0157] Imposing the values of inductances to be equal to the values L₁*,L₂*, L₃* and L₄* respectively, gives four relations (L₁* values aretransformed values)

M ₁=(L _(p1*) /L _(p1))^(1/2)

M ₄=(L _(s4) */L _(s4))^(1/2)

[0158] It is required to make the second resonator in form b definedabove with reference to FIG. 2b.

L _(sb2) *=m ₂ ² M _(pa 2)[1+C _(pa2) /C _(sa2v)]²

m ₂ =[L _(sb2)/(M_(pa2)(1+C _(pa2) /C _(sa 2v)))]

L _(a) =M _(pa 2) ; C ₂ =C _(pa 2) ; C ₃ =C _(sa 2)

[0159] It is required to make the third resonator without changing theform (form b defined above).

M ₃=(L _(sb 3) */L _(sb 3))^(1/2)

[0160] The form selected for the filter imposes that m₂<m₃; the 2N−1degrees of freedom (in this case 7 degrees of freedom) are used to fixthe four values of inductances and the values of the three capacitances(for example C_(sa 2 w) and C_(pb 3 w) and one more). Obviously multiplechoices are possible, for example including the choice made in the nextexample. It would also be possible for some capacitances (three maximumin this case if the inductances are fixed) to have defined valuescorresponding to usual standardized values.

[0161] We will now give a first numeric example illustrating the processaccording to the invention.

[0162] We will consider the order 8 zigzag prototype in FIG. 9. For thetransformation, we chose a value of the inductance equal to 20 nH thatgives approximately the best overvoltage that could be hoped for at thecentral frequency and a satisfactory self-resonant frequency. We choseto fix the ratios of the capacitances of the two three-elementresonators equal to 20, and to keep their form equal to the form of theprototype circuit, and finally we chose the value of capacitanceC_(sv b) (namely C_(sb v)=0.5 pF before transformation), which givessatisfactory values for elements of the filter. In this case, therecannot be more than one solution. The calculation shows that a solutionactually exists (it satisfies inequality conditions for transformationratios, and the values of all components are positive).

[0163] The transformed filter circuit is shown in FIG. 10 and the valuesof the elements on Table 2 below. An analysis is carried out to makesure that the response obtained with this circuit really is actuallyidentical to the response obtained for the prototype (except possibly atfrequencies far from the pass band where the response is slightlyimproved by the addition of matching components—in this case—for lowfrequencies close to zero).

[0164]FIG. 11 shows responses of the prototype filter and thetransformed filter using the process according to this invention. TABLE2 Value of elements of the transformed circuit in $ Transformed filterfile 1.CKT ATNB Topology RS = 75.00 1 RL = 75.00 10 1 CAP Cs 1 20.23302E−12 2 IND La 2 0 0.20000E−07 3 CAP Ca 2 0 0.54639E−12 4 CAP Cst2 3 0.80812E−12 5 CAP Cpt 3 0 0.27808E−10 6 CAP Cp2 3 5 0.32113E−10 7IND Ls2 3 4 0.20000E−07 8 CAP Cs2 4 5 0.16056E−11 9 CAP Cpt 5 00.39559E−10 10 CAP Cst 5 7 0.13480E−08 11 CAP Cp3 7 0 0.32338E−10 12 INDLs3 7 6 0.20000E−07 13 CAP Cs3 6 0 0.16169E−11 14 CAP Cst 7 80.11259E−11 15 CAP Cpt 8 0 0.77261E−12 16 CAP Csb 8 9 0.22989E−10 17 INDLsb 9 10 0.20000E−07 18 CAP Cpa 10 0 0.15858E−10

[0165] The main advantage of the process according to the invention isthat it can take account of all possibilities and can be generalized toa significant number of filter topologies.

[0166] It was mentioned above that it was applicable to the case ofzigzag filters, polynomial LC pass band filters in either of thepossible dual forms (obviously the systematic transformation principleremains the same, but it is much simpler to implement because there areonly arms with two elements).

[0167] More generally, the analysis of the transformation process showsthat it is applicable to ladder filters comprising not more than oneinductance per arm (or that can be transformed to satisfy thiscondition) such that a pair of transformers can be inserted for eachinductance and one of these transformers can be eliminated afterdisplacement by a Norton transformation. This assumes that there are twocapacitances (one in series in a horizontal arm and one in parallel in avertical arm) derived from the separation of capacitances in the armscontaining resonators with two or three elements, or that alreadyexisted in the circuit (series or shunt capacitance) and that arelocated in the direction of displacement of the transformer after theinductance to be transformed and before the next inductance (so that allinductances can be transformed independently).

[0168] This is done for filters that have a topology such that thefollowing two conditions can be satisfied, with the possible exceptionof one or more dipoles located at the end of the filter:

[0169] a series capacitance can be extracted from the horizontal armscontaining an inductance and a shunt capacitance can be extracted fromthe vertical arm along the direction of displacement of the transformer,

[0170] and/or

[0171] a shunt capacitance can be extracted from the vertical armscontaining an inductance and a series capacitance can be extracted fromthe next horizontal arm (along the direction of displacement of thetransformer).

[0172] The condition is also satisfied if one of the previous twoconditions is replaced by the condition in which there is an existingcapacitances Π or T after any dipole containing a single inductance.

[0173] This is illustrated in FIG. 12 that shows sub-circuits usingvarious current topologies.

[0174] Obviously, the process that has just been described with respectto some specific embodiments can be generalised.

[0175] Moreover, transformations (Colin, Saal and Ulbrich, etc.) areknown for taking filters containing dipoles with more than oneinductance in some of the arms or bridges, and making them equivalent tocases with a ladder topology to which the process described can beapplied.

[0176] We will describe some other examples in the following.

[0177] As we saw in the previous example, for a given filter topologycontaining three-element resonators, there is a possible choice betweentwo different forms in the transformation, for each of these resonators.For example, for zigzag filters containing a two-element resonator ateach end, and if N is the order of the filter and K is the number ofthree-element resonators of the filter (we have K=N/2−2), there willthen be 2^(k)=2^((N/2−2)) possible alternative configurations for thetransformed filter. Moreover, these different configurations can easilybe taken into account in a calculation program and in the following wewill describe the advantage of taking them into consideration, withreference to the examples given.

[0178] It is also possible to take account in the calculation ofdifferent variants for narrow band matching at the ends of the filters,which in some cases is attractive, since the added matching components(inductance or capacitance) modify the behaviour of the filter input andoutput impedances outside the pass band in different ways, but that canbe very favourable for some applications (for example for makingduplexers, the choice between modifying the response at zero frequencyand at infinite frequency may be important).

[0179] Note that narrow band matching leads to very satisfactorysolutions for pass bands significantly wider than what are normallycalled narrow pass bands. In practice, as a result of this no limitationin performances was observed in any of the cases dealt with (namely forrelative pass bands of up to about 30%). It was mentioned above thatother matching techniques may be used for wider relative bands.

[0180] In the equation for which the principle was described above,inequalities appear that have to be respected, together with non-linearrelations between parameters. Since positive real values are necessaryfor the elements, acceptable solutions for the different possiblealternative configurations for the transformed filter can only existwithin the range in which available parameters can vary (value ofinductances, etc.). The possible range of “good” values of inductances(achievable with good characteristics) is sufficient in practice toalways obtain solutions without it being necessary to include values ofcapacitances that are difficult to obtain (too small).

[0181] The degrees of freedom added into the proposed transformationprocess in practice can be used to make the filter with all inductancesbeing equal, which is economically the most attractive case. Theremaining degrees of freedom may be used in different ways. We havedescribed above the advantage of choosing off-the-shelf values forcapacitances. Another particularly interesting advantage is to fix theratios of capacitances in three-element resonators, since this is ameans of making filters using piezo-electric filters with bulk orsurface waves. This advantage of the process will be described furtherbelow.

[0182] The ratio of capacitances in three-element resonators (L₁ C₁ C₀)also has attractive fundamental properties; this is a simple function ofthe relative separation of remarkable frequencies of these dipoles, andconsequently it has a strong influence on the maximum pass band of afilter using these resonators. It is easy to show that, in thetransformations made, this parameter has the important property that itcan only decrease. Therefore, after transformation for eachthree-element resonator, its only possible values lie within theinterval between zero and its value in the prototype circuit.

[0183] 2. We will now describe other examples of high order filters,beginning with the transformation of order 12 zigzag ellipticalprototype.

[0184] This example is given to illustrate possibilities of the processmentioned above and particularly to demonstrate the advantage of takingaccount of all possible configurations of transformed filters (using twoalternative topologies of three-element resonators). It is observed thatfor a transformation in which a single value of inductances is to beobtained, the range in which solutions exist as a function of otherparameters (for example capacitance ratios) is much wider if allpossible configurations for filters are authorised.

[0185] This example relates to the transformation according to theproposed process for an order 12 zigzag filter calculated to have a passband varying from 51.0 to 59.0 MHz with 0.3 dB ripple and attenuatedband rejection of 70 dB. The circuit for the prototype filter consideredis shown in FIG. 13.

[0186] The value of the elements of the prototype filter is shown intable 3 below. TABLE 3 Zigzag filter order = 12 RS = 88.62 1 RL = 75.009 1 CAP Ca 1 0 0.28120E−09 2 IND La 1 0 0.29156E−07 3 CAP C2s 1 20.11249E−10 4 IND L2p 2 3 0.21206E−06 5 CAP C2p 2 3 0.27711E−10 6 CAPC3s 3 4 0.51800E−10 7 IND L3s 4 0 0.23281E−06 8 CAP C3p 3 0 0.11827E−099 CAP C4s 3 5 0.13630E−10 10 IND L4p 5 6 0.13534E−06 11 CAP C4p 5 60.47714E−10 12 CAP C5s 6 7 0.20679E−10 13 IND L5s 7 0 0.53070E−06 14 CAPC5p 6 0 0.77829E−10 15 CAP Cb 6 8 0.45671E−11 16 IND Lb 8 9 0.19189E−05

[0187] The objective is to use a single value for all inductances and itwas decided to fix the value of the ratio of capacitances ofthree-element resonators to 1 (which gives equal values). In this case,only one parameter remains free. It was decided that this free parameterwould be the capacitance C_(sb v)=V_(s 4 v) which is determined bydividing the capacitances of the series LC resonator at end B of thefilter. Therefore, there can only be a simple infinite number ofsolutions for a given topology. It is found that the allowable solutionsdo not contain all different possible configurations (in this case 16)at the same time, depending on the common value for all inductances anddepending on the values of the other parameters. Thus, for example, inthe chosen case (inductance=500 nH, capacitance ratios=1 and C_(sb v)varying within a small range above the minimum possible value), thesolutions found belong to only four topologies. Other configurations arepossible for other values of parameters (capacitance ratio).

[0188] Tables 4, 5, 6 and 7 below show four solutions corresponding tofour different configurations, as examples. The corresponding circuitsfor the transformed filters are given in FIGS. 14, 15, 16 and 17respectively. It is found that the values of the elements aresatisfactory in three cases, while in the fourth case (denoted ATNTTB),the value of one capacitance (C_(a)) is too low to accommodate thedistributed capacitance of the inductance that it matches. If there weregood reasons for using this configuration, the values of some parametersshould be modified to obtain a more favourable circuit.

[0189] Note that in the following, the configuration is identified bythe form of the three-element resonators, in comparison with theirconfiguration in the prototype zigzag filter. A represents end A, Brepresents end B and the letter T indicates that the form of thethree-element resonator is transformed from its form in the prototype,while N means that its form has not been transformed. By convention, tofacilitate the division of capacitances, in the prototypes we willconsider that the resonators in the horizontal arms are in form (a) inFIG. 2 and that the resonators in the vertical arms are in form (b) inthe same figure. Thus, the ATNTNB configuration is composed ofthree-element resonators all with form (b) in FIG. 2 (composed of aseries LC resonating arm in parallel with a capacitance (called thestatic capacitance)). For example, this topology is useful to complywith standard practice in representing piezoelectric resonators when itis required to create filters based on these elements (capacitance andinductance values “equivalent” to resonators given by resonatormanufacturers correspond to this form). TABLE 4 Filter in FIG. 4Elements of the transformed filter with ATNTNB topology (file 2.CKT) RS= 75.00 1 RL = 75.00 15 1 CAP Cs 1 2 0.88144E−11 2 IND La 2 00.50000E−06 3 CAP Ca 2 0 0.10364E−11 4 CAP Cst 2 3 0.85705E−11 5 CAP Cpt3 0 0.37657E−10 6 CAP Cp2 3 5 0.23505E−10 7 IND Ls2 3 4 0.50000E−06 8CAP Cs2 4 5 0.23505E−10 9 CAP Cpt 5 0 0.50139E−10 10 CAP Cst 5 70.55173E−10 11 CAP Cp3 7 0 0.24119E−10 12 IND Ls3 7 6 0.50000E−06 13 CAPCs3 6 0 0.24119E−10 14 CAP Cst 7 8 0.13601E−10 15 CAP Cpt 8 00.71387E−11 16 CAP Cp4 8 10 0.25830E−10 17 IND Ls4 8 9 0.50000E−06 18CAP Cs4 9 10 0.25830E−10 19 CAP Cpt 10 0 0.52705E−10 20 CAP Cst 10 120.52838E−08 21 CAP Cp5 12 0 0.21948E−10 22 IND Ls5 12 11 0.50000E−06 23CAP Cs5 11 0 0.21948E−10 24 CAP Cst 12 13 0.17875E−10 25 CAP Cpt 13 00.16115E−10 26 CAP Csb 13 14 0.61406E−10 27 IND Lsb 14 15 0.50000E−06 28CAP Cpa 15 0 0.65170E−10

[0190] TABLE 5 Filter in FIG. 15 Elements of the transformed filter withANTTNB topology file 7.Ckt RS = 75.00 1 RL = 75.00 15 1 CAP Cs 1 20.88144E−11 2 IND La 2 0 0.50000E−06 3 CAP Ca 2 0 0.29189E−11 4 CAP Cst2 3 0.81040E−11 5 CAP Cpt 3 0 0.13752E−10 6 CAP Cs2 3 4 0.11725E−10 7IND Lp2 4 5 0.50000E−06 8 CAP Cp2 4 5 0.11752E−10 9 CAP Cpt 5 00.60453E−11 10 CAP Cst 5 7 0.66522E−11 11 CAP Cs3 7 6 0.12060E−10 12 INDLp3 6 0 0.50000E−06 13 CAP Cp3 6 0 0.12060E−10 14 CAP Cst 7 80.67999E−11 15 CAP Cpt 8 0 0.13938E−10 16 CAP Cp4 8 10 0.25830E−10 17IND Ls4 8 9 0.50000E−06 18 CAP Cs4 9 10 0.25830E−10 19 CAP Cpt 10 00.52702E−10 20 CAP Cst 10 12 0.53837E−08 21 CAP Cp5 12 0 0.21948E−10 22IND Ls5 12 11 0.50000E−06 23 CAP Cs5 11 0 0.21948E−10 24 CAP Cst 12 130.15793E−10 25 CAP Cpt 13 0 0.14238E−10 26 CAP Csb 13 14 0.80596E−10 27IND Lsb 14 15 0.50000E−06 28 CAP Cpa 15 0 0.65170E−10

[0191] TABLE 6 Filter in FIG. 16 Elements of the transformed filter withANTNTB topology file 11.CKT RS = 75.00 1 RL = 75.00 15 1 CAP Cs 1 20.88144E−11 2 IND La 2 0 0.50000E−06 3 CAP Ca 2 0 0.53213E−11 4 CAP Cst2 3 0.42858E−11 5 CAP Cpt 3 0 0.72725E−11 6 CAP Cs2 3 4 0.11752E−10 7IND Lp2 4 5 0.50000E−06 8 CAP Cp2 4 5 0.11752E−10 9 CAP Cpt 5 00.12531E−10 10 CAP Cst 5 7 0.13789E−10 11 CAP Cs3 7 6 0.12060E−10 12 INDLp3 6 0 0.50000E−06 13 CAP Cp3 6 0 0.12060E−10 14 CAP Cst 7 80.34054E−11 15 CAP Cpt 8 0 0.17874E−11 16 CAP Cs4 8 9 0.12915E−10 17 INDLp4 9 10 0.50000E−06 18 CAP Cp4 9 10 0.12951E−10 19 CAP Cpt 10 00.94377E−11 20 CAP Cst 10 12 0.94615E−09 21 CAP Cs5 12 11 0.10974E−10 22IND Lp5 11 0 0.50000E−06 23 CAP Cp5 11 0 0.10974E−10 24 CAP Cst 12 130.78967E−11 25 CAP Cpt 13 0 0.22135E−10 26 CAP Csb 13 14 0.80596E−10 27IND Lsb 14 15 0.50000E−06 28 CAP Cpa 15 0 0.65170E−10

[0192] TABLE 7 Filter in FIG. 17 Elements of the transformed filter withATNTTB topology file 14.ckt RS = 75.00 1 RL = 75.00 15 1 CAP Cs 1 20.88144E−11 2 IND La 2 0 0.50000E−06 3 CAP Ca 2 0 0.17609E−11 ** seecomment 4 CAP Cst 2 3 0.96266E−11 5 CAP Cpt 3 0 0.42297E−10 6 CAP Cp2 35 0.23505E−10 7 IND Ls2 3 4 0.50000E−06 8 CAP Cp2 4 5 0.23505E−10 9 CAPCpt 5 0 0.40114E−10 10 CAP Cst 5 7 0.44141E−10 11 CAP Cs3 7 00.24119E−10 12 IND Ls3 7 6 0.50000E−06 13 CAP Cs3 6 0 0.24119E−10 14 CAPCst 7 8 0.28860E−10 15 CAP Cpt 8 0 0.15148E−10 16 CAP Cp2 8 100.25830E−10 17 IND Ls2 8 9 0.50000E−06 18 CAP Cs2 9 10 0.25830E−10 19CAP Cpt 10 0 0.19662E−10 20 CAP Cst 10 12 0.19277E−10 21 CAP Cs3 12 110.10974E−10 22 IND Lp3 11 0 0.50000E−06 23 CAP Cp3 11 0 0.10974E−10 24CAP Cst 12 13 0.73687E−11 25 CAP Cpt 13 0 0.20655E−10 26 CAP Csb 13 140.99785E−10 27 IND Lsb 14 15 0.50000E−06 28 CAP Cpa 15 0 0.65170E−10

[0193] Responses calculated for the prototype filter and the differenttransformed filters are identical near the pass band as can be seen byexamining FIGS. 18 and 18b, which illustrate the response of theprototype filter and transformed filters (lossless) and details of theresponse of the prototype filter and the transformed filters in the passband (the curves are precisely superposed) (FIGS. 18a and 18 b below).Components added for narrow band matching slightly improve the responseat very low frequencies in the above case.

[0194] 3. We will now describe a variant of the process by which anumber of standard capacitances can be used.

[0195] During the previous presentation of the principle of the processaccording to the invention, it was mentioned that after fixing the valueof the n inductances, there were 2n−1 degrees of freedom. These degreesof freedom can be used to fix the value of some capacitors. This wouldmake it possible to choose off-the-shelf standard models for particularcapacitances (frequently low values for which the choice is morerestricted and the price is higher for a given precision), at therequired precision (typically from 1 to several % depending on the orderof the filter), and manufacturers naturally have all required values.

[0196] Several techniques are possible for solving this question thatmust combine the global solution of the previous equations with aproblem similar to a problem with integer numbers. The inventors chose aprocess that takes place in two steps. In the first step, in choosingsome auxiliary parameters (the ratios of the capacitances ofthree-element resonators and one of the capacitances derived from thedecomposition of the one in the two-element resonator at end B), therange in which a solution exists, and particularly the rangecorresponding to capacitance values that are neither too small or toolarge, is determined for the chosen value(s) of the inductances. Thesecond step is to limit variations of parameters to within this range,and then to start from one end of the filter (end B), to solve theequations given above by choosing at least one of the impedancetranslation capacitances located between the resonators according to analgorithm using the value calculated at the centre of the range, from atable of standard values (E24, etc.). The result is thus at least asmany standard capacitances as the number of additional degrees offreedom available.

[0197] 4. We will now describe application of the process in the case offilters with TEM dielectric resonators.

[0198] Remember that dipoles with two discrete LC elements in series orin parallel or with three discrete elements (L, C₁, C₂) of zigzagfilters may be made with a very good approximation using TEM dielectricresonators and capacitances (usually either zero or only one). Thecoaxial resonator may be terminated in open circuit or in short circuitand it can be used either close to one of its “series” resonances(resonance Z=0) or one of its “parallel” resonances (anti-resonanceY=0). The addition of a shunt capacitance or a series capacitance willresult in electrical behaviour very similar to the behaviour of a dipoleresonator with three discrete elements, within a given frequencyinterval.

[0199] Dielectric resonators are characterised by their length or by oneof their resonant frequencies (theoretical) at no load (inverselyproportional to the length), these magnitudes being adjustable bymechanical means (or by the addition of adjustable discrete components)and by their characteristic impedance Z_(car) that is determined by thegeometric characteristics of the resonator and the dielectricpermittivity of the material used.

[0200] Technologically, the characteristic impedance is usually between6 Ohms and 16 Ohms depending on the relative permittivity of thematerial and the straight section of the bar, but other slightly higheror slightly lower values can be made in special cases.

[0201] Therefore, the degrees of freedom will be used, under theconstraint of a characteristic impedance, to identify impedances of theresonator alone or the resonator and its coupling elements (seriesand/or shunt capacitances) so as to obtain a good approximation of theimpedance of two or three element resonators. For example, in order toachieve this, we will identify:

[0202] the characteristic frequencies at which Z=0 or Z=∞

[0203] the derivatives ∂z/∂ω and/or ∂(1/Z)/∂ω at these same pointsrespectively; or

[0204] Z and its derivative ∂Z/∂ω at the centre of the pass band and theanti-resonant frequency at which Z=∞.

[0205] A practical example showing application of the method in thiscase is described below.

[0206] We will use an LC prototype with the same topology as that in thefirst example given above for transformation of LC filters (FIG. 4). Inthis case the filter is narrower, and was synthesised to have infiniteattenuation points at specified frequencies close to the pass band.

[0207] The circuit for the prototype filter is illustrated in FIG. 19,and its characteristics and the values of elements will be given with anexample of a transformed circuit.

[0208] We assume that the transformation principle indicated for LCfilters is known. Starting from this step, we will identify the fourresonators (two with two elements and two with three elements) withcoaxial TEM resonator structures. The identification process is knownand it is usually not unique, it depends on the choice made for theresonator type (λ/2, λ/4, etc.) and also on the frequency area in whichit is desirable to maximise the precision of using a dipole containing adistributed element to approximately represent the impedance(admittance) of the resonator dipole with discrete elements.

[0209] The following illustrates examples of some of the possiblevariants in the cases encountered.

[0210] LC resonators are treated as coaxial resonators with λ/4 or λ/2lines.

[0211] It will be noted that this invention may use at least one circuitin which some of the resonators are made by using the dielectricresonator in series, with one of the accesses being on the central coreof the resonator and the other access of the quadripole being on theexternal metallisation of the resonator, rather then being in shunt (indipole, with the other end of the resonator either being grounded or inopen circuit).

[0212] The first 2-element resonator illustrated in FIG. 19 may betransformed as illustrated in FIG. 20.

[0213] Thus, the parallel resonator L_(p1)-C_(p1) in FIG. 19 may betransformed either into a parallel resonator in λ/4 in short circuit ona vertical branch, or into a parallel resonator in λ/2 in open circuiton a vertical branch.

[0214] The resonant frequency of the residual shunt LC circuit {L_(p1)C_(p1v)} is sufficiently above the resonance of {L_(p1) C_(p1)}.Strictly speaking, it would be necessary to identify the middle ω₀ ofthe pass band of admittance values and its derivative, firstly for theL_(p1) C_(p1v) form, and secondly for production with a coaxialresonator. For simplification purposes, the anti-resonant frequency f₁(Z=∞ or Y=0) and the slope ∂(1/Z)/∂ω at this point, will be identified.These approximate values can then be specified better, if necessary, byvarying the anti-resonant frequency of the coaxial resonator.

j(C _(p1v) /L _(p1))^(1/2) m ₁ ⁻²(f/f ₁ −f ₁ /f)≅2jC _(p1v) /L_(p1))^(1/2) m ₁ ⁻²(f−f ₁)/f ₁

[0215] For a λ/4 resonator in anti-resonant short circuit at f₁:

(jZ _(car) tan(πf/2f ₁))⁻¹ ≅jπ(f−f ₁)/(2f ₁ Z _(car))

[0216] therefore:

C _(p1v) =m ₁ ⁴ L _(p1)(π/(2A ₁ Z _(car)))²

[0217] (Where A₁=2 for λ/4 and A₁=1 for short circuit λ/2

C _(p1w) =C _(p1)−(Rω ₀)⁻¹(m ₁ ²−1)^(1/2) −C _(p1v)>0

[0218] gives an upper limit to the value of m₁ unless Z_(car) is higher.

[0219] As a variant, an approximate resolution by identification ofvalues and derivatives in f₀ at the centre of the pass band, givesC_(p1v) and the frequency of f₁+ε of the coaxial resonator.

[0220] The second resonator illustrated in FIG. 19 (with three elementslocated in an horizontal arm) may be transformed as illustrated in FIG.21.

[0221] Thus the resonator in FIG. 19 comprising a capacitance C_(s 2 v)in series with a circuit comprising an inductance M_(p a 2) in paralleland a capacitance C_(p a 2) may be transformed into either a seriesresonator on a horizontal arm in λ/4 in short circuit associated with aseries capacitance C_(c2), or a series resonator on a horizontal branchin λ/2 in open circuit associated with a capacitance C_(c2).

[0222] The infinite attenuation (anti-resonant) frequency ω_(a2) must beconserved, the resonant frequency ω_(b2) is a priori in the pass bandand the impedance slope must be conserved at this point. This is done byidentifying ∂(Z)/∂ω in ω_(b2) starting from the responses of equivalentcircuits with discrete elements and resonator.

[0223] The impedance of the k/2 line dipole and capacitance is:

Z=−j/C _(c2) jZ _(car) Cot (πω/ω _(a2)).

[0224] The identification at ω_(b2) (Z=0) gives:

C _(c2)=−tan(πω_(b2)/πω_(a2))/(Z _(car)ω_(b2))=−π tan(q)/(Z _(car)ω_(a2)q)

[0225] where q=πω_(b2)/ω_(a2)

[0226] For a λ/4 resonator in anti-resonant short circuit at ω_(a2), theresult is C_(c2)=1/(Z_(car)ω_(b2) tan(q/2)).

[0227] The slope at Ob2 may be identified as follows.

[0228] For the shape with discrete elements:

∂(Z)/∂ω=2jL _(sb 2 equiv)=2jm ² M _(pa2)(1+C _(pa2) /C _(sa2v))

[0229] at the series resonant frequency ω_(b2) (f_(2b)), where Z=0.

[0230] L_(sb 2 equiv) is the inductance of the three-element resonatorin form b in FIG. 2.

[0231] The value of ∂(Z)/∂ω at ω_(b2) to make the coaxial resonator is:$\frac{\partial(z)}{\partial\omega} = {{{- {jz}_{car}}\left\{ {\frac{\cot\left( {\pi \quad {\omega_{b2}/\omega_{a2}}} \right.}{\omega_{b2}} - \frac{{{\pi csc}\left( {\pi \quad {\omega_{b2}/\omega_{a2}}} \right)}^{2}}{\omega_{a2}}} \right\}} = {{- j}\quad z_{car}\left\{ {\frac{\pi \quad {\cot (q)}}{q\quad \omega_{a2}} - \frac{{{qcsc}(q)}^{2}}{\omega_{a2}}} \right\}}}$

[0232] The relation between q and the capacitances ratio is:$\frac{C_{sa2v}}{C_{pa2}} = \frac{\left( {\pi^{2} - q^{2}} \right.}{q2}$

[0233] The result is then:$\frac{\partial Z}{\partial\omega} = {{2\quad j\quad m_{2}^{2}{M_{pa2}\left( {1 + {C_{pa2}/C_{sa2v}}} \right)}^{2}} = {2\quad {jm}_{2}^{2}M_{pa2}\frac{\pi^{4}}{\left( {\pi^{2} - q^{2}} \right)^{2}}}}$

[0234] and the value of m₂ is obtained by identifying the two values of∂(Z)/∂ω at ω_(b2) for the λ/2 line circuit and the discrete circuit.$m_{2}^{2} = \frac{\left( {\pi^{2} - q^{2}} \right)^{2}{Z_{car}\left\lbrack {{q\quad {\csc (q)}^{2}} - {\cot (q)}} \right\rbrack}}{2M_{pa2}\pi^{3}q\quad \omega_{a2}}$

[0235] which can be developed close to q=π to give$m_{2}^{2} = {\frac{2Z_{car}}{M_{pa2}\pi \quad \omega_{a2}} + \frac{\left( {1 + {4{\pi^{2}/3}}} \right){Z_{car}\left( {q - \pi} \right)}^{2}}{{2 \cdot M_{pa2}}\pi^{3}\omega_{a2}} + {o\left\lbrack \left( {q - \pi} \right)^{3} \right\rbrack}}$

[0236] It can be seen that m₂ could be obtained as a function of thecapacitances ratio, and using this magnitude as a free parameter gives asimple numeric solution without any approximation; this is the choicethat was actually used in the following examples.

[0237] With a λ/4 resonator in short circuit, the result would be:$m_{2}^{2} = \frac{\left( {\pi^{2} - q^{2}} \right)^{2}{Z_{car}\left\lbrack {q + {\sin (q)}} \right\rbrack}{\sec^{2}\left( {q/2} \right)}}{{4 \cdot M_{pa2}}\pi^{3}q\quad \omega_{a2}}$

[0238] and approximately (development around π/2):$m_{2}^{2} = {\frac{4\quad Z_{car}}{M_{pa2}\pi \quad \omega_{a2}} + \frac{\left( {{4/\pi} + {4{\pi/3}}} \right){Z_{car}\left( {q - \pi} \right)}^{2}}{{4 \cdot M_{pa2}}\pi^{2}\quad \omega_{a2}} + {o\left\lbrack \left( {q - \pi} \right)^{3} \right\rbrack}}$

[0239] Starting from the relation expressing the decomposition of thecapacitance C_(sa2) into three parts and considering the Nortontransformation already made between resonators 1 and 2, the result is:

C _(sa2w) ⁻¹ =C _(sa2) ⁻¹ −C _(sa2v) ⁻¹ −C _(p1w) ⁻¹(m ₁ /m ²⁻1) since C_(sa2u) =C _(p1w)(m ₁ /m ²⁻1)⁻¹

[0240] The third resonator illustrated in FIG. 19 (three elements in avertical arm) may be transformed as illustrated in FIG. 22.

[0241] Thus, the resonator in FIG. 19 comprising a capacitanceC_(pb 3 v) in parallel with a branch formed from an inductance Ls inseries with a capacitance C_(s) may be transformed either into aparallel resonator on a vertical branch at λ/4 in short circuit inseries with a capacitance C_(c 3), or into a parallel resonator on avertical branch at λ/2 in open circuit in series with a capacitanceC_(c 3).

[0242] For the third resonator, the infinite attenuation pointcorresponds to Z=0 at ω_(b3) that has to be conserved, whereas inprinciple ω_(a3) is in pass band and ∂(1/Z)/∂ω will be identified atω_(a3).

[0243] The condition for an infinite attenuation point at f_(b3) gives:

C _(c3)=−tan(πω_(b3)/ω_(a3))/(2Z _(car)ω_(b3))

[0244] The derivative should be identified at ω_(a3):

[0245] For the circuit with discrete components:

∂(1/Z)/∂ω=2jm ₃ ⁻² C _(pa3)=2jm ₃ ⁻² C _(pb3v)(1+C_(pb3v) /C _(sb3))

[0246] For the circuit with half wave line in open circuit:

∂(1/Z)/∂ω=jπ/(Z _(car)ω_(a3))

[0247] Let ω_(a3)=(1+t)ω_(b3) since f_(a3) is free whereas f_(b3) isfixed in advance; f_(a3) and C_(pb3v) are related; (it will be observedthat t is a simple function of the ratio of the capacitances(1+t)²=(1+C_(sb3)/C_(pb3v)))

ω_(b3) ²=ω_(a3) ²/(1+C _(sb3) /C _(pb3v))=(1+t)²ω_(b3) ²/(1+C _(sb3) /C_(pb3v)),

[0248] therefore: C_(pb3v)=C_(sb3)/(t(2+t)) and

m ₃ ²=(2Z _(car)ω_(b3) C _(sb3)/π)(1+t)³/(t ²(2+t)²)

[0249] The conditions that limit the choice of t are:

m ₃ >m ₂ therefore t<t _(max)=(Z _(car3)ω_(b3) C _(sb3)/(2π))^(1/2) /m ₂

C _(pb3w)>0 or C _(pb3v) <C _(pb3) or t>t _(min) =C _(sb3)/(2C _(pb3))

[0250] (denoting Z_(car3) the characteristic impedance of the coaxialresonator used for the third resonator). If t_(min)>t_(max), resonatorswith different characteristic impedances have to be used.

[0251] It is observed that m₂ varies with (Z_(car2))^(1/2), hence:

C _(sb3)/(2C _(pb3))<t _(min) <t<t _(max)=[(Z _(car3)/(8Z _(car2)))C_(sb3)ω_(b3) M _(pa2)ω_(a3)]^(1/2)

[0252] The last resonator illustrated in FIG. 19 (horizontal arm seriesLC resonator) may be transformed as illustrated in FIG. 23.

[0253] Thus, the resonator in FIG. 19 comprising a capacitance C_(s4) inseries with an inductance L_(s4) may be transformed either in a seriesresonator on a horizontal branch in λ/4 in short circuit with an inputon the central core and output on the external metallisation of theresonator, or into a series resonator on a horizontal branch in λ/2 inopen circuit, or a series resonator on a horizontal branch in λ/4 inshort circuit in series with an inductance L_(s 4 w).

[0254] For example, the last resonator {L_(s4) m₄ ⁻², C_(s4v) m₄ ⁻²} maybe made by a residual inductance associated with a resonator centredapproximately on the pass band (FIG. 24d); L_(s4)=L_(s4v) 30 L_(s4w);L_(s4v) corresponds to the equivalent inductance of the dielectricresonator.

[0255] Then: L_(s4v) C_(s4v)=L_(s4) C_(s4)=1/ω² ₄

[0256] By identifying$2\quad j\quad m_{4}^{2}\sqrt{\frac{L_{sv4}}{C_{sv4}} = {j\quad {Z_{car}/A_{4}}}}$

[0257] Hence

m ² ₄ =C _(s4v) Z _(car)ω₄π/2A ₄

[0258] And/or C_(s4v)=(2A ₄ m ² ₄)/(ΠZ _(car)ω₄)

[0259] where A₄=1 for a half-wave resonator in open circuit and

[0260] A₄=2 for a quarter-wave resonator.

[0261] Other techniques may also be considered, such as the technique ofincluding the series narrow band matching inductance (positive ornegative) in end B in the transformed inductance, or including theseries narrow band matching inductance (positive or negative) in thecapacitance C_(s4b) (then C_(s4w)=0). In all cases, a more complexcalculation can be used to precisely identify the centre of the passband on a computer.

[0262] The solution obtained by combining the various expressions isexpressed as follows as a function of the q and t parameters:

C _(p1v) =m ₁ ⁴ L _(p1)(π/(2A ₁ Z _(car)))²

[0263] where A₁=2 for a quarter wave and 1 for a half wave

C _(p1w) =C _(p1)−(R_(ω) ₀))⁻¹(m ₁ ²−1) ^(1/2) −C _(p1v)

[0264] This limits the possible values of m₁, except to increaseZ_(car1), q close to π determines the following for a half wave:

C _(sa2v) =C _(pa2)[−1+(q/π) ²]

C _(sa2w) ⁻¹ =C _(sa2) ⁻¹ −C _(sa2v) −C _(p1w) ⁻¹(m ₁ /m ₂−1)

[0265] whereC_(sb3)/(2C_(pb3))<t_(min)<t<t_(max)=[(Z_(car3)/(8Z_(car2)))C_(sb3)ω_(b3)M_(pa2)ω_(a3)]^(1/2)

M ₃ ²=(2Z _(car)ω_(b3) C _(sb3)/π)(1+t)³/(t ²(2+t)²)

C _(pb3v) =C _(sb3)/(t(2+t))

C _(pb3w) =C _(pb3) −C _(sa2w)(m ₃ /m ₂−1)−C _(pb3v)

[0266] m₄ is determined by comparing:

C _(s4u) =C _(pb3w)/(m ₃ /m ₄−1)

[0267] and

C _(s4u) ⁻¹ =C _(s4) ⁻¹ −πZ _(car4)ω₄/(2A ₄ m ₄ ²)−(Rω ₀)(1−m ₄ ² N²)^(1/2) /m ₄

[0268] which gives an equation in 1/m₄ written as follows:

(m ₃ /m ₄−1)C _(pb3w) =C _(s4) ⁻¹−(πZ _(car4)ω₄ /A ₄)/m ₄ ²−(Rω ₀)(1/m ₄² −n ²)^(1/2)

[0269] or

(πZ _(car4)ω₄ /A ₄)/m ₄ ²+(Rω ₀)(1/m ₄ ² −n ²)^(1/2) +m ₃ C _(pb3w) ⁻¹/m ₄ =C _(pb3w) ⁻¹ +C _(s4)

[0270] hence

C _(s4v)=(2A ₄ m ₄ ²)/(πω₄ Z _(car4))

[0271] then

L _(s4v)=1/(C _(s4v)ω₄ ²)

L _(s4w) =L _(s4) −L _(s4v)

[0272] and,

C_(s12) =C _(sa2u)/(m ₁ m ₂) C_(p12) =C _(p1w)/(m ₁ m ₂)

C _(p23)=(1/m ₂−1/m ₃)C_(sa2w) C _(s23) =C _(sa2w)/(m ₂ m ₃)

C _(s34) =C _(pb3w)/(m ₃ m ₄(m ₃ /m ₄−1)) C _(p34) =C _(pb3w)/(m ₃ m ₄)

C _(p4) =root(1/m ₄ ² −n ²)/(Rω ₀ n)

[0273] The data given below could provide a numeric solution for thiscase.

[0274] The prototype is a filter with the same topology as that shown inFIG. 19, but this is a synthesised filter to give specified infiniteattenuation frequencies (888 MHz and 899 MHz) very close to the passband that varies from 890 to 897 MHz. The ripple in the pass band is 0.2dB. Attenuated band rejections (not equal in this case) are of the orderof slightly less than 30 dB. The following table contains values ofelements in the equivalent prototype circuit. TABLE 8 Rs = 75.00 R1 =75 * n²; n² = 0.645; 1 Cap CP1 = Ca 1 0 4.3800E−10 2 Ind Lp1 = La 1 07.2200E−11 3 Cap Csa2 1 2 1.6750E−11 4 Cap Cpa2 2 3 1.7030E−10 5 IndLpa2 2 3 1.8400E−10 6 Cap Csb3 3 4 8.0600E−14 7 Ind Lsb3 4 0 3.9860E−078 Cap Cpb3 3 0 8.1350E−12 9 Cap Cs4 = Csb 3 5 3.1000E−12 10 Ind Ls4 =Lsb 5 6 1.0284E−06

[0275] We chose capacitances matching (positive and negative) at eachend and as mentioned above, the decomposition of the inductance of theseries resonator at end B (+1 parameter) into two (L_(sbu)+L_(sbv)). 4parameters are fixed by fixing the characteristic impedance and theequivalence, and there are still 3+1 parameters available (theadditional parameter is derived from decomposition of the inductanceL_(sb)). It was decided that the negative matching capacitance at end Awill cancel the transformed capacitance Cp1u, that the quarter wave lineat end B will resonate at the central frequency and that the negativematching capacitances at end B will cancel the transformed capacitanceC_(s4w). There is still one free parameter that may be equal to thevalue of an inductance or of a capacitor. The value chosen is C_(s4v).

[0276]FIG. 24 shows an example solution thus obtained.

[0277] Table 9 below illustrates the values of elements in the TEM linescircuit for a particular solution. TABLE 9 Terminations Rs = 75.00 1 Rl= 75.00 15 1 Cap 1 2 c1 = Cs1 = 0.407d−12 2 LGN 2 0 Line 1 lambda/4 cc,zc = 11.5, fal = 922.195d6 3 Cap 2 3 c2 = cs12 = 0.602d−12 4 Cap 3 0 c3= cp12 = 0.740d−12 5 Cap 3 4 c4 = cc2 = 0.649d−12 6 LGN 4 5 Line 2lambda/2 co, zc = 11.5 Ohm, fa2 = 899.000d6 7 Cap 5 0 c5 = cp23 =0.00583d−12 8 Cap 5 6 c6 = cs23 = 0.512d−12 9 Cap 5 7 c7 = cc3 =0.523d−12 10 LGN 7 0 Line 3 lambda/2 co, zc = 11.5, fa3 = 897.590d6 11Cap 7 8 c8 = cs34 = 0.625d−12 12 Cap 8 0 c9 = cp34 = 22.1d−12 13 LGN 8 9Line 4 lambda/4 co, zc = 11.5, fr = 891.370d6 14 Ind 9 10 11 = Lbw 3.263nH 15 Cap 10 11 c11 = cp4 49.77d−12

[0278] An analysis was carried out to verify that the transformationgives a good result. FIG. 25 shows the response calculated with roundedvalues of capacitances for lines with equal characteristic impedance.

[0279] 5. We will now describe a second example application for order 12elliptical filters.

[0280] This example applies to transformation of an order 12 zigzag LCelliptical prototype with a central frequency of 520 MHz with a passband of 40 MHz with a ripple of 0.3 dB and an attenuated band rejectionof 70 dB (see the circuit shown in FIG. 26 and the response given inFIG. 27). At this frequency, the high inductances of the prototypecircuit are difficult to make (self resonance too low) and secondlydielectric resonators enable an appreciable performance gain,particularly in terms of the loss of insertion due to their highovervoltage. The principle described above for order 8 can easily beextended to higher orders by iteration on pairs of three-elementresonators.

[0281] In the transformation, the parameters we chose were the n−2=4capacitance ratios of three-element resonators (in the final transformedstate) and one capacitance C_(s4v)=C_(sbv). The three-element resonatorsare always transformed to obtain an open circuit half wave line inseries with a capacitance. As before, we also chose quarter waves inshort circuit and in open circuit for ends A and B respectively. In thiscase we will only use one additional matching component at each end (theother end being absorbed in a component of the transformed circuit),which implies additional relations and reduces the number of degrees offreedom.

[0282] For narrow band matching at the ends of the filter, the formulasdescribed above are applied with the possibility of choosing theadditional matching component (inductance or capacitance) at each end.The first case may be useful particularly for making multiplexers.

[0283] As for the case of filters based on the LC technology, theprogram determines the range of existence of solutions (by the variationof C_(sbv) within its possible variation range and by variation of thefour capacitance ratios between their initial values and about {fraction(1/20)} of the initial values) (avoiding values too close to zero (whichis the minimum allowed) that result in capacitance values that are tooextreme). By definition, a solution exists for given values ofparameters when all inequalities mentioned above are respected and whenall calculated values of the filter elements are positive.

[0284] In the example chosen, 17 steps were used for each of thecapacitance ratios, and 10 steps were used for C_(sbv). The calculationshows about 100 000 solutions out of about 835 000 testedconfigurations.

[0285] The next step was to calculate about a hundred solutions withparameters distributed within the existence range, for each matchingcase at ends A and B and in the case of two typical values of thecharacteristic impedance (11.5 ohms and 7.0 ohms). Solutions werechecked by analysis.

[0286] The following is an example determination of the solutions space(for matching by inductance at both ends).

[0287] Table 10 below contains the extreme values of parameters leadingto solutions: TRANSFORMATION OF THE order 12 ZIGZAG ELLIPTICAL PASS BAND(bp6_520) CALCULATIONS FOR Zc = 11.50000 Ohms PRELIMINARY STEP, No.calculations = 835 210 No. solutions = 111206 Existence of solutions forparameters in the following ranges: (Csbv and resonator capacitanceratios working from A towards B) C_(sbvmin) = 0.49666D−12 C_(sbvmax) =0.24828D−11 RAPC2_(min)(1) = 0.24063D+00 RAPC2_(max)(1) = 0.36629D+01RAPC3_(min)(1) = 0.83876D+00 RAPC3_(max)(1) = 0.26688D+01 RAPC2_(min)(2)= 0.34033D+00 RAPC2_(max)(2) = 0.51805D+01 RAPC3_(min)(2) = 0.16288D+01RAPC3_(max)(2) = 0.41712D+01

[0288] In fact, in this case, the existence range is more complicated(fragmented) than is apparent looking at these simplified data. In thecase of matching at the ends by two capacitances or by an inductance anda capacitance, it is observed that the existence range remains fairlysimilar to the existence range indicated above. One of the advantages ofthis analysis of the range in which solutions exist is that it showsthat it is also possible to obtain a circuit using five judiciouslypredetermined values of capacitances in the final circuit (for examplestandard series values) and to provide a systematic method for obtainingsolutions of this type.

[0289] Solutions can then very quickly be calculated for a restrictedrange of variation of parameters. Numeric solutions obtained at variouspoints within the range in which solutions exist for two matching casesat the ends are given below.

[0290] Numeric examples of solutions: transformation of an order 12filter to input lines with Zc=11.5 Ohms.

[0291] The characteristics of the target filter are as follows:

[0292] Central frequency 520 MHz,

[0293] Pass band 40 MHZ,

[0294] Better than 15 dB attenuation at F<490 MHz and F>550 MHz+−,

[0295] Ripple in pass band 0.3 dB,

[0296] Rejection in attenuated band 80 dB (equi-ripple).

[0297] The circuit for the prototype filter with elliptical responsethat satisfies the required envelope curve is shown in FIG. 26 (order 12zigzag prototype filter).

[0298] Table 11 below gives the values of elements of the zigzagprototype filter circuit illustrated in FIG. 26 denormalised) (MKSAunits). TABLE 11 RS = 85.20 1 RL = 75.00 15 1 CAP Ca 1 0 0.59486E−10 2IND La 1 0 0.15580E−08 3 CAP Cs1 1 2 0.94744E−12 4 IND Lp1 2 30.19285E−07 5 CAP Cp1 2 3 0.38757E−11 6 CAP Cs2 3 4 0.43877E−11 7 INDLs2 4 0 0.26838E−07 8 CAP Cp2 3 0 0.17063E−10 9 CAP Cs3 3 5 0.11324E−1110 IND Lp3 5 6 0.12138E−07 11 CAP Cp3 5 6 0.65513E−11 12 CAP Cs4 6 70.17677E−11 13 IND Ls4 7 0 0.62614E−07 14 CAP Cp4 6 0 0.10744E−10 15 CAPCb 6 8 0.24827E−12 16 IND Lb 8 9 0.38478E−06

[0299] The calculated response of the zigzag prototype filter is givenin FIG. 27.

[0300] We will now give a first example transformation: matching by twoinductances.

[0301] The transformation is done as described above. The parametersused are the ratios of the resonator capacitances and when the range ofexistence of solutions has been determined, a solution is chosen thatgives acceptable values of elements located within this existence rangefor a low value of the Csbv parameter. The advantage of includinginductances at the ends of the filter is considerable. Firstly, theyhelp to significantly reduce undesirable responses in attenuated banddue to the multi-mode nature of the resonators. Secondly, their lossesdo not influence the response of the filter if care is taken to takethem into account in source and load resistances.

[0302] The LC equivalent transformed circuit is shown in FIG. 28, andtable 12 below gives values of elements (LC equivalent elements): TABLE12 (File LL58.CKT ellipt. Zigzag bp6_520) RS = 75.00 1 RL = 75.00 15 1IND Lsa 1 2 0.34662E−07 2 IND La 2 0 0.44940E−08 3 CAP Ca 2 00.20961E−10 4 CAP Cst1 2 3 0.19033E−11 5 CAP Cpt1 3 0 0.81304E−11 6 CAPCs2 3 4 0.15559E−09 7 IND Lp2 4 5 0.20015E−08 8 CAP Cp2 4 5 0.37343E−109 CAP Cpt2 5 0 0.87980E−10 10 CAP Cst2 5 6 0.20161E−09 11 CAP Cs3 6 70.37702E−10 12 IND Lp3 7 0 0.16975E−08 13 CAP Cp3 7 0 0.31670E−10 14 CAPCst3 6 8 0.11219E−10 15 CAP Cpt3 8 0 0.13692E−11 16 CAP Cs2 8 90.17119E−10 17 IND Lp2 9 10 0.20645E−08 18 CAP Cp2 9 10 0.38547E−10 19CAP Cpt4 10 0 0.21913E−10 20 CAP Cst4 10 11 0.59035E−10 21 CAP Cs3 11 120.17435E−10 22 IND Lp3 12 0 0.20129E−08 23 CAP Cp3 12 0 0.37554E−10 24CAP Cst5 11 13 0.72521E−11 25 CAP Cpt5 13 0 0.26855E−10 26 CAP Csb 13 140.34582E−10 27 IND Lsb 14 15 0.28211E−08 28 ind Lpb 15 0 0.27824E−08

[0303] Table 13 below gives the values of parameters used for atransformed lines circuit (the ratios of the capacitances ofthree-element resonators and transformation ratios, the free parameterC_(sbv) is chosen to be equal to 5×10⁻¹² F): TABLE 13 RAPC2(1) =0.24000D+00 RAPC3(1) = 0.84000D+00 RAPC2(2) = 0.22500D+01 RAPC3(2) =0.21540D+01 (capacitance ratio of 3-element dipoles (transformationparameters) FILTER CIRCUIT WITH DIELECTRIC RESONATORS step csbv = 1 CSBV= 0.50000D−12 Transformation ratio Na = 0.106658D+01 M1 = Ma =0.16984D+01 SOURCE = 75 000 Ohms Ls-H matching = 0.346624D−07 Verticalarm Lp-Cp Lpeq = 0.449397D−08 cpeq = 0.209611D−10 OR 1/4 line O. SC Zc =0.115000D+02 Faa = 0.518559D+09 Impedance translation: Cs-H(1, 2) =0.190326D−11 Cp-V(1, 2) = 0.813040D−11 Transformation ratio used M2 =0.32216D+00 Horizontal parallel resonator Fr = 0.256112778D+09 Fa =0.582152101D+09 Csr2 = 0.155595D−09 Lpr2 = 0.200153D−08 Cpr2 =0.373427D−10 OR 1/2 LINE O. OC at Fa, Zc = 0.115000D+02 Cseries = Cspr2= 0.155595D−09 Impedance translation: Cp-V(2, 3) = 0.879795D−10 Cs-H(2,3) = 0.201608D−09 Transformation ratio used M3 = 0.46275D+00 Verticalseries resonator fr = 0.463796316d+09 fa = 0.686430486d+09 Csr3 =0.204903D−10 Lsr3 = 0.574695D−08 Cpr3 = 0.172119D−10 or Verticalparallel resonator fr = 0.463796316D+09 fa = 0.686430486D+09 Lpr3 =0.169747D−08 Cpr3 = 0.316698D−10 Csr3 = 0.377022D−10 OR 1/2 LINE O. O.C.at Fa, Zc = 0.115000D+02 Cseries = Cspr3 = 0.377022D−10 Impedancetranslation: Cs_H(3, 4) = 0.112193D−10 Cp_V(3, 4) = 0.136919D−11Transformation ratio used M4 = 0.41242D+00 Horizontal parallel resonatorFr = 0.469611479+09 Fa = 0.564402756D+09 Csr4 = 0.171187D−10 Lpr4 =0.206447D−08 Cpr4 = 0.385171D−10 OR 1/2 LINE O. O.C. at Fa, Zc0.115000D+02 Cseries = Cspr4 = 0.171187D−10 Impedance translation:Cp_V(4, 5) = 0.219132D−10 Cs_H(4, 5) = 0.590349D−10 Transformation ratioused M5 = 0.565550D+00 Vertical series resonator fr = 0.478381789D+09 fa= 0.578872121D+09 Csr5 = 0.2552780D−11 Lsr5 = 0.200234D−07 Cpr5 =0.119069D−10 Or vertical parallel resonator fr = 0.478381789D+09 fa =0.578872121D+09 Lpr5 = 0.201287D−08 Cpr5 = 0.375543D−10 Csr5 =0.174347D−10 OR 1/2 LINE O. O.C. at Fa, Zc = 0.115000D+02 Cseries =Cspr5 = 0.174347D−10 Impedance translation: Cs_H(5, 6) = 0.725212D−11Cp_V(5, 6) = 0.268545D−10 Transformation ratio used M6 = Mb =0.12024D+00 No. = 0.10000D+01 LC series arm Leq = 0.282114−08 Ceq =0.345820D−10 Fr = 0.509545251D+09 OR 1/4 LINE O. O.C. Zc = 0.115000D+02Fr = 0.509545251D+09 in series Capa-H Csbrest = 0.100000D+31 Self //lpbdapt = 0.27842D−08 Termination = 75 000 Ohms END OF TRANSFORMEDCIRCUIT, step of csbv no = 1 Result of calculations written to file =LL58.CKT

[0304] A simulation is made to check that the response obtained with thetransformed circuit is actually the same as the response of theprototype circuit. The response of the transformed filter with matchingby two inductances is shown in FIG. 29 (it will be noted that theaddition of inductive matching components and/or the use of distributedresonators slightly improves the response outside the band (the effectincreases as the distance from the pass band increases).

[0305] We will now give a second example of transformation of the sameprototype filter but with matching of two capacitances at both ends Aand B.

[0306] The procedure is the same and values of parameters were chosenfairly arbitrarily. The form of the transformed circuit (LC equivalent)is the same as the form of the circuit given for the previous example,except for the use of two capacitances at both ends, and in this casethe values of the elements are given in table 14 below: TABLE 14 RS =75.00 1 RL = 75.00 15 1 CAP Csa 1 2 0.28978E−11 2 IND La 2 0 0.40955E−083 CAP Ca 2 0 0.19103E−10 4 CAP Cst1 2 3 0.19959E−11 5 CAP Cpt1 3 00.80490E−11 6 CAP Cs1 3 4 0.15559E−09 7 IND Lp1 4 5 0.20015E−08 8 CAPCp1 4 5 0.37343E−10 9 CAP Cpt2 5 0 0.85527E−10 10 CAP Cst2 5 60.19530E−09 11 CAP Cs2 6 7 0.37702E−10 12 IND Lp2 7 0 0.16975E−08 13 CAPCp2 7 0 0.31670E−10 14 CAP Cst3 6 8 0.28836E−10 15 CAP Cpt3 8 00.35192E−11 16 CAP Cs3 8 9 0.17119E−10 17 IND Lp3 9 10 0.20645E−08 18CAP Cp3 9 10 0.38517E−10 19 CAP Cpt4 10 0 0.61695E−11 20 CAP Cst4 10 110.10255E−10 21 CAP Cs4 11 12 0.14707E−10 22 IND Lp4 12 0 0.20732E−08 23CAP Cp4 12 0 0.38681E−10 24 CAP Cst5 11 13 0.12380E−10 25 CAP Cpt5 13 00.55386E−10 26 CAP Csb 13 14 0.68678E−10 27 IND Lsb 14 15 0.56027E−08 28CAP Cpb 15 0 0.33597E−10

[0307] Table 15 below contains the value of elements in the transformedlines circuit and the values of parameters used in the globaltransformation: TABLE 15 Iteration no = 37 RAPC2(1) = 0.24000D+00RAPC3(1) = 0.84000D+00 RAPC4(2) = 0.22500D+01 RAPC5(2) = 0.26300D+01(capacitance ratios for the three-element dipoles chosen for thetransformation) FILTER CIRCUIT WITH DIELECTRIC RESONATORS step csbv no =2 CSBV = 0.10000D−11 Transformation ratio Na = 0.106658D+01 M1 = Ma =0.16213D+01 SOURCE = 75 000 Ohms Cs-H matching = 0.289776D−11 Verticalarm Lp-Cp Lpeq = 0.409550D−08 Cpeq = 0.191025D−10 OR 1/4 line 0. SC Zc =0.115000D+02 Faa = 0.569012D+09 Impedance translation: Cs-H(1, 2) =0.199589D−11 Cp-V(1, 2) = 0.804901D−11 Transformation ratio used M2 =0.32216D+00 Horizontal parallel resonator Fr = 0.256112778D+09 Fa =0.582152101D+09 Csr2 = 0.155595D−09 Lpr2 = 0.200153D−08 Cpr2 =0.373427D−10 OR LINE 1/2 O. OC at Fa, Zc = 0.115000D+02 Cseries = Cspr2= 0.155595D−09 Impedance translation: Cp-V(2, 3) = 0.852272D−10 Cs-H(2,3) = 0.195301D−09 Transformation ratio used M3 = 0.46275D+00 Verticalseries resonator fr = 0.463796316D+09 fa = 0.686430486D+09 Csr3 =0.204903D−10 Lsr3 = 0.574695D−08 Cpr3 = 0.172119D−10 Or verticalparallel resonator fr = 0.463796316D+09 fa = 0.686430486D+09 Lpr3 =0.169747D−08 Cpr3 = 0.316698D−10 Csr3 = 0.377022D−10 OR 1/2 LINE O. O.C.at Fa, Zc = 0.115000D+02 Cseries = Cspr3 = 0.377022D−10 Impedancetranslation: Cs_H(3, 4) = 0.288363D−10 Cp_V(3, 4) = 0.351915D−11Transformation ratio used M4 = 0.41242D+00 Horizontal parallel resonatorFr = 0.469611479+09 fa = 0.564402756D+09 Csr4 = 0.171187D−10 Lpr4 =0.206447D−08 Cpr4 = 0.385171D−10 OR 1/2 LINE O. O.C. at Fa, Zc =0.115000D+02 Cseries = Cspr4 = 0.171187D−10 Impedance translation:Cp_V(4, 5) = 0.616955D−11 Cs_H(4, 5) = 0.102549D−10 Transformation ratioused M5 = 0.66053D+00 Vertical series resonator fr = 0.478381789D+09 fa= 0.562017808D+09 Csr5 = 0.405163D−11 Lsr5 = 0.273188D−07 Cpr5 =0.106558D−10 Or vertical parallel resonator fr = 0.478381789D+09 fa =0.562017808D+09 Lpr5 = 0.207323D−08 Cpr5 = 0.386805D−10 Csr5 =0.147074D−10 OR 1/2 LINE O. O.C. at Fa, Zc = 0.115000D+02 Cseries =Cspr5 = 0.147074D−10 Impedance translation: Cs_H(5, 6) = 0.123796D−11Cp_V(5, 6) = 0.553864 D−10 Transformation ratio used M6 = Mb =0.12067D+00 No. = 0.10000D+01 LC series arm Leq = 0.560267D−08 Ceq =0.686783D−10 Fr = 0.256574124D+09 OR 1/4 LINE O. O.C. Zc = 0.115000D+02Fr = 0.256574124D+09 Matching capacitance // Cpbdapt = 0.335971D−10Termination = 75 000 Ohms END OF TRANSFORMED CIRCUIT, step of csbv no =2 Result of calculations written to file = CC43.CKT

[0308] The simulation to check the response of the transformed filterwith matching by capacitances was made, and this response is shown inFIGS. 30a and 30 b for the pass band and the entire responserespectively. There is an extremely small modification to the responseoutside the band due to the use of TEM resonators and/or matching(improvement at the high frequencies end).

[0309] Another application of this invention is for the design ofmatchable filters.

[0310] Available degrees of freedom (either all degrees of freedom, ordegrees of freedom remaining after a transformation has been made toobtain given inductances or particular resonators) may (also) be used toreduce the number of variable capacitances in matchable filters, forexample for which it is desired to keep the pass band approximatelyconstant over the entire matching range. Thus, the result is thatcoupling quadripoles between resonators can be kept constant, or theirvariation may be limited.

[0311] 6. We will now describe application of the process according tothis invention to crystal resonator filters.

[0312] Crystal piezoelectric resonators have unequalled overvoltagecharacteristics (up to several million and typically several hundredthousand for quartz) and stability characteristics as a function of thetemperature and time, so that high performance filters with narrow orvery narrow relative bands can be made. They use electromechanical wavesthat propagate within the volume or on the surface of monocrystallinesolids (quartz, lithium tantalate, lithium niobate, langasite (lanthanumsilico-gallate), gallium phosphate, etc.) or piezoelectric ceramics(lead titano-zirconate, etc.). They are distributed multi-modecomponents (like dielectric resonators) with an electrical behaviourthat is described approximately by an equivalent three-element resonatortype circuit [15 to 19], near one of their electromechanical resonances(there is also an infinite number of resonances). Approximatetransmission line models of crystal resonators are also known. But thesemodels are too complicated to be used for the synthesis of filters(several lines). Equivalent circuits with discrete elements comprisingseveral series resonant arms are also more frequently used, particularlyto analyse the effects of other modes after a synthesis.

[0313] Diagrammatically, existing technologies can be used to make bulkwave resonators within the 10 kHz-10 GHz frequency range and surfacewave resonators within the 10 MHz-10 GHz range (resonant frequencies arehighly complex functions of parameters characterising the material andgeometric parameters characterising resonators). The dependence of theseresonators is frequently dominated by the dependence of one of theparameters (the thickness for many bulk wave resonators, and the pitchof interdigitised combs for surface wave resonators).

[0314] As mentioned above, these resonators are characterised by a ratioof capacitances fixed approximately by the material, the wave type andthe crystalline orientation (for example of the order of 0.52% for theslow shear bulk wave of cut quartz AT), and furthermore values ofinductances that can be achieved for a given frequency under “goodconditions” (in terms of performance, undesirable responses and cost),are located within a usually very narrow range of values (for exampletowards the centre of the 5.7 mH to 10.9 mH range for AT quartz at 20MHz). Since the filters are very narrow, the “most easily achievable”values of equivalent inductances of the different resonators are veryclose (close frequencies), and we can often force them to be equalwithout much difficulty.

[0315] Bulk wave resonators operating at frequencies of severalGigaHertz can be obtained using thin layers of piezoelectric materials(ZnO, AIN, TaLiO₃, etc.). It has been shown that thin crystalline layersof AIN epitaxied on gallium arsenide can be made so that filtering andamplification functions can be achieved. This technology has been knownfor a long time, but is difficult to control and can be used to makevery miniaturised filters that can be co-integrated withmicroelectronics. Existing applications are aimed at the 500 MHz-3000MHz frequency range. It has been determined that the upper frequencylimit is from 10 to 30 GHz depending on the materials.

[0316] The principles of the different types of surface wave resonatorsand their properties are briefly mentioned below. Filters using thesedevices are now becoming very important due to the advantage of reducingthe volume and cost of components in equipment, and particularly insubscriber terminals. For example, they are extremely small (a few mm³)at the frequencies used in earth and satellite personal radiocommunications, and it would be possible to envisage making filter banksor integrated multiplexers on the same substrate, so as to makemultistandard and multi-frequency terminals.

[0317] A crystal filter can be made using the process according to theinvention based on the following principle.

[0318] Diagrammatically, the principle is to use degrees of freedomintroduced in the general transformation method to identify resonatorswith a given inductance and a given capacitances ratio. One of theresonant or anti-resonant frequencies is free. The other is imposed bythe position of infinite points.

[0319] Going into more detail, a number of refinements can beimplemented, like the following. If the precise dependence of the ratioof the capacitance and inductance on the different parameterscharacterising the resonator (that can be calculated by digital models[19]) is known, it is possible to determine the most favourable range(narrower than the previous range) of values of inductances achievablefor each resonant frequency from the point of view of their performancesand cost (moreover, the inventors have demonstrated that there arevalues of parameters for crystal resonators for filters that optimisethe electrical characteristics, and also almost optimise the cost ofthese resonators at the same time [18]). Within this narrower range ofinductance values, the precise value of the capacitances of theequivalent circuit and their ratio is obtained (this ratio is thenalmost independent of all parameters characterising the resonators(insensitivity) and it remains close to the usual value characterisingthe material, the given crystalline orientation and the wave type used),while the sensitivity of the inductance to parameters other than thethickness is practically zero (reduction of manufacturing dispersions).Furthermore, it is generally possible to force their design such thatthe only difference in the manufacture of different resonators(different frequencies) is in a parameter that is easy to control (forexample the thickness of the crystalline blade or the thickness ofmetallisation). Thus, the end result is that it is possible toeconomically make filters with higher performance responses (resultingfrom the choice of prototypes with minimum inductance with rationalresponses) than are possible with polynomial responses usually used forcrystal filters.

[0320] We will now give an example embodiment of a bulk wave resonatorfilter using the process according to the invention.

[0321] We will now consider an intermediate frequency filter with a 1.5MHz pass band, centred at 70 MHz and with a high shape factor (3 dB/60dB transition in less than 500 kHz) and 400 ohm termination impedances(equal). We chose a degree 12 elliptical response which, with fourfinite infinite attenuation frequencies generated by crystal resonatorsand close to the pass band (at 0.68162046D+08 Hz, 0.68641759D+08 Hz,0.71879273D+08 Hz; 0.71376924D+08 Hz), provides the required stiffnessof the transition band.

[0322] The material chosen is lithium tantalate with a cut very close toX that gives resonators with a much greater theoretical capacitanceratio than quartz (C_(p)/C_(s)#8k²/n²π²+ε(k⁴) where k # 44% comparedwith # 8% for quartz. With this material, inductances of the order of40.0 μH can be obtained for resonant frequencies close to 68.5 MHz and adynamic capacitance Cs of the order of 0.135 pF. In practice, theexperimental static capacitance is of the order of 1.5 pF. It includes aglobal parasite capacitance of the order of 0.65 pF, itself comprisingcapacitances representing the package and a very low capacitance betweenaccesses. The circuit of the filter takes account of these parasitecapacitances. In order to simplify the presentation, we preferred totake them into account globally and consider the total “static”capacitance (which corresponds to the case in which the package is leftdisconnected). The overvoltage of these resonators is of the order of2500. Furthermore, we would like to use two inductances of the order of400 nH at the ends of the filter, to create helical resonators in smallcavities with a minimum overvoltage of 350-400. However, the overvoltageof these elements is not very critical in this case, and it can bedecided to use inductances on a very small torus of carbonyl ironpowder.

[0323]FIG. 31 shows the equivalent circuit for a lithium tantalateresonator with fr # 68.5 MHz.

[0324] As can be seen in FIG. 31, the resonator can be considered like acapacitance C₀′ placed in parallel with a branch comprising acapacitance C_(s) and an inductance L_(s) in series.

[0325] The resonator can also be considered like a circuit comprisingthe components that have just been described, and also two additionalcapacitances C_(p1) and C_(p2) that connect the circuit input and outputrespectively to a capacitance C_(p0) between the input and output.

[0326] When the metal package of the resonator is left disconnected, thetwo circuits are equivalent with C′₀=C₀+C_(p0)+(C_(p1−C)_(p2))/(C_(p1)+C_(p2))

[0327] where C_(p0), C_(p1), C_(p2) represent the parasite capacitancesbetween the input and output and between these accesses and the package.

[0328]FIG. 32 shows the circuit for the initial degree 12 zigzagelliptical prototype. The transformation is made by imposing acapacitances ratio of 11.2=C′₀(total)/C_(s) and an inductance of 40 μH.The only free variable is C_(sbv), that will be chosen to have a lowfrequency resonance. In this case, the transformation process has theadvantage that it enables production of filters taking account ofparasite capacitances of crystals in the case in which the package is“grounded” (with some additional calculations). This possibility isuseful if it is required to obtain relative bandwidths close to half thetheoretical value of the capacitances ratio. The transformation processalso has the advantage that it is the only process for making filterswith a global response using crystals without using balancedtransformers.

[0329] The values of elements of the equivalent order 12 zigzag filtercircuit illustrated in FIG. 32 are given in table 16 below. TABLE 16 RS= 493.66 1 RL = 400.00 9 1 CAP Ca 1 0 0.27519E−09 2 IND La 1 00.18703E−07 3 CAP Cs1 1 2 0.13778E−11 4 IND Ls1 2 3 0.18211E−06 5 CAPCs1 2 3 0.26921E−10 6 CAP Cs2 3 4 0.10866E−11 7 IND Ls2 4 0 0.50177E−058 CAP Cp2 3 0 0.19739E−10 9 CAP Cs3 3 5 0.17563E−11 10 IND Lp3 5 60.10651E−06 11 CAP Cp3 5 6 0.46679E−10 12 CAP Cs4 6 7 0.43679E−12 13 INDLs4 7 0 0.12308E−04 14 CAP Cp4 6 0 0.12527E−10 15 CAP Cb 6 8 0.95337E−1316 IND Lb 8 9 0.54583E−04

[0330] The circuit for the transformed filter to include lithiumtantalate crystal resonators is shown in FIG. 33. For the centralfrequency, the characteristics of TaLiO3 resonators with cut X are asgiven above. The same value was chosen for the two inductances at theends, namely L 400 μH.

[0331] Table 17 below shows the values of elements in the equivalentcircuit after transformation (Topology for ATNTNB crystals). TABLE 17 RS= 400.00 1 RL = 400.00 15 1 CAP Cs 1 2 0.11280E−11 2 IND La 2 00.40000E−06 0.350E+03|Fr(LC) = 84.379 MHz 3 CAP Ca 2 0 0.88941E−11 4 CAPCst 2 3 0.39169E−11 5 CAP Cpt 3 0 0.10994E−10 6 CAP Cp1 3 5 0.14953E−117 IND Ls1 3 4 0.40000E−04 0.250E+04|Fr(crystal) = 68.870500 MHz 8 CAPCs1 4 5 0.13351E−12 9 CAP Cpt 5 0 0.14607E−11 10 CAP Cst 5 7 0.11030E−1111 CAP Cp2 7 0 0.15266E−11 12 IND Ls2 7 6 0.40000E−040250E+04|Fr(crystal) = 68.161982 MHz 13 CAP Cs2 6 0 0.13630E−12 14 CAPCs2 7 8 0.73416E−12 15 CAP Cpt 8 0 0.57081E−12 16 CAP Cp3 8 100.15164E−11 17 IND Ls3 8 9 0.40000E−04 0250E+04|Fr(crystal) = 68.388142MHz 18 CAP Cs3 9 10 0.13540E−12 19 CAP Cpt 10 0 0.18325E−11 20 CAP Cst10 12 0.13582E−10 21 CAP Cp4 12 0 0.15053E−11 22 IND Ls4 12 110.40000E−04 0250E+04|Fr(crystal) = 68.161982 MHz 23 CAP Cs4 11 00.13440E−12 24 CAP Cst 12 13 0.77126E−12 25 CAP Cpt 13 0 0.15471E−10 26CAP Csb 13 14 0.34115E−08 27 IND Lsb 14 15 0.40000E−06 0350E+03|Fr(LC) =43.084 MHz 28 CAP Cpb 15 0 0.66159E−10

[0332]FIG. 34 shows the response of the transformed crystals filtercalculated without losses. FIGS. 35a and 35 b show the response of thetransformed filter considering losses. The overvoltages considered areas indicated in the above table (after the values of inductances). It isobserved that the process provides a filter with an excellent shapefactor with few resonators, that in practice only differ by a differentfinal frequency adjustment (step which is in any case essential andautomated).

[0333] We will now describe an example application of the invention forsurface wave filters.

[0334] Developments show a simple example of how the transformationprocess proposed according to the invention can be applied to obtainhigh frequency filters involving surface wave resonators underconditions facilitating achieving and/or increasing performances. Someuseful variants are then discussed, which are equally applicable to thecase of bulk wave filters, and particularly to cases of resonators usingthin piezoelectric layers.

[0335] The operating principle of surface wave resonators has been knownfor more than 25 years. But the most important developments are recentand are still ongoing in some cases. This is mainly due to the fact thattheir most important uses are related to the development of mobile radiocommunications. In this domain, ongoing efforts towards miniaturisationof user terminals facilitate their use, including in multiband.

[0336] In their “dipole” variant considered here, they are composed of acentral interdigitised transducer located between two networkscomprising a large number of fingers that reflect the acoustic energyemitted by the transducer in the two opposite propagation directions,and thus confine it in the central region of the resonator. Anotherapproximately equivalent structure often used is composed of a narrowtransducer with a very large number of fingers. This structure usesreflections internal to the transducer that are very significant onhighly piezoelectric materials.

[0337] In the direction transverse to the propagation, confinement isalso (usually) obtained due to discontinuities in the velocity caused bythe presence of “rails” bringing in the current surrounding thetransducers (and possibly the reflecting networks) and a free space onthe outside. The principle and the method of calculating these devicesare described in recent literature (for example see references [16 and22]). Various surface wave types are currently used in the devices. Themost frequently used are Rayleigh waves (pure or generalised),transverse or quasi-transverse waves (SSBW, STW) and longitudinalsurface pseudo-waves (L-SAW). The latter two wave types areschematically almost bulk waves that are more or less guided (held)close to the surface by modifications to the surface (corrugation ormetallisation) that modify their velocity locally. Their decay withinthe thickness of the wafer may be slower than the decay (exponential) ofRayleigh waves (which is an advantage when making filters resisting agiven power since for equal power it reduces the acoustic energydensity).

[0338] In general, and very schematically, transverse wave resonatorsproduce better overvoltages than resonators using Rayleigh waves. Theresult is close to the intrinsic value for the materials (centralfrequency). Remember that the overvoltage varies with 1/Fc, such thatfor example, Q_(intrinsic.)Fc=constant # 1.5×10¹³ for quartz andprobably # 2×10¹³ for lithium tantalate. Usually, overvoltages forRayleigh wave devices are also better than for devices usinglongitudinal pseudo-surface waves (these waves are intrinsicallyslightly dissipative).

[0339] The equivalent circuit usually considered for these devices isthe same as the circuit indicated for bulk wave resonators (see FIG.31).

[0340] The following is an example application of the process accordingto the invention to an order 8 filter:

[0341] It is required to use two identical resonators involving theslightly dissipative quasi-longitudinal surface mode (L-SAW) of lithiumtantalate with cut Y+42° and two inductances with the same value.

[0342] The central frequency of this filter is fixed at 900 MHz, and itmust be terminated on 50 Ohms and have a pass band of 20 MHz (ripple 0.3dB), an attenuation of 20 dB at + and −5 MHz from the corners of thepass band and 30 dB starting from ±7.5 MHz from the pass band. Theprototype circuit chosen to satisfy this envelope curve consists of anorder 8 zigzag elliptical filter (same type of circuit as in FIG. 19),and the values of the elements of the initial circuit are indicated intable 18 below. It can be seen that the different inductances in thisfilter are very different and that making the circuit directly resultsin very serious difficulties, since firstly the orders of magnitude ofthe two end inductances are very different from the values that arepossible with acceptable characteristics, and secondly equivalentcircuits for the two three-element resonators are very different fromeach other and none of them corresponds well to the values that can beobtained for surface wave resonators using a known material and wavetype under good response and overvoltage conditions. TABLE 18 Values ofelements of the prototype filter circuit ## zigzag Filter order = 8 No.bp900x_4 ## Fc 900 MHz Bw = 20 MHz, Rip. = 3 dB Rej. = 35 Db RS = 65.121 RL = 50.00 6 1 CAP Ca 1 0 0.14050E−09 2 IND La 1 0 0.22100E−09 3 CAPCs1 1 2 0.12145E−11 4 IND L2 2 3 0.89762E−09 5 CAP C2 2 3 0.33419E−10 6CAP C3 3 4 0.28741E−12 7 IND L3 4 0 0.11346E−06 8 CAP C4 3 0 0.79086E−119 CAP Cb 3 5 0.68362E−13 10 IND Lb 5 6 0.46082E−06

[0343] This circuit is transformed according to the principles mentionedabove, such that the two three-element resonators are made in the formof two identical longitudinal surface quasi-wave resonators that areslightly dissipative and that the two-element resonators at the ends aremade using inductances. This arrangement has the advantage that theattenuation poles and most of the attenuation zeros are made usingsurface wave resonators (high overvoltage) and also that inductance andcapacitance resonators are used, thus contributing to eliminatingundesirable off-band responses of surface wave resonators.

[0344] Close to the chosen central frequency, it is known how to makethese resonators in the form of a long transducer with a large number offingers (slightly more than a hundred) on the lithium tantalate with aY+48° or Y+52° cut. The values of other manufacturing parameters(aperture, metallisation value, etc.) can be chosen to give a goodresponse, a good overvoltage (700-800) and to be extremely compact.Under these conditions, these resonators are characterised at thecentral frequency by an equivalent inductance equal to 104.5 nH, aglobal static capacitance close to about 3.6 nF, a coupling coefficientof about 10.57% and a capacitance ratio C₀/C₁=11.696.

[0345] It was also chosen to use two equal inductances with a low value(20 nH) that can easily be made inside the ceramic package used for thefilter (for example in the form of spirals obtained by silk screenprinting on an inside face of this miniature package), and such thatthey have a fairly high overvoltage.

[0346] The circuit obtained after the transformation proposed above isillustrated in FIG. 36.

[0347] The parameters used during the transformation and the values ofthe elements obtained after the transformation are given in table 19below. TABLE 19 ***GLOBAL TRANSFORMATION PARAMETERS Calculation forC0/C1 = 0, 11695D+02 Csbv varies from 0.200D−11 to 0.220D−10, 5 stepsCalculation step No. 2 parameter Csbv = 7.000 D−12 Topology: ANNBeliminated (inequalities) Topology: ATNB (L-SAW) selected in advance(inequalities) Topology: ANTB eliminated (inequalities) Topology: ATTBnot selected in advance (Piezo. Res.) For topology No. 2 ATNBTransformation ratio Na = 0.11413D+01; M1 = 0.95131D+01 Transformationratio M 2 = 084992D+00 Transformation ratio M 3 = 0.95972D+00Transformation ratio M 4 = 0., 0833D+00 Nb = 0.10000D+01 ***VALUES OFCOMPONENTS OF THE TRANSFORMED FILTER Calculation No = 2 End terminationa = 50.000 Cs-H matching = 0.327172D−12 Cp-V matching eliminated =0.324397D−12 LC arm // Vertical Lat = 0.2000D−07 Cat = 0.973991D−12 fa =0.114032258D+10 Impedance translation: Cs-H 1- 2 = 0.279058D−12 Cp-V 1-2 = 0.284440D−11 L-SAW resonator horizontal arm fr = 0.881987102D+09 fa= 0.918921629D+09 Lsr 2 = 0.104500D−06 Csr 2 = 0.311601D−12 Cpr 2 =0.364418D−11 Impedance translation: Cp-V 2_3 = 0.523883D−11 Cs-H 2_3 =0.405534D−10 L-SAW resonator vertical arm fr = 0.881359165D+09 fa =0.91826739D+09 Lsr 3 = 0.104500D−06 Csr 3 = 0.312046D−12 Cpr 3 =0.364937D−11 Impedance translation: Cs_H 3- 4 = 0.380127D−12 Cp_V 3- 4 =0.137103D−11 LC arm series horizontal Lbt = 0.200D−07 Cbt = 0.16129D−09fr = 0.886143154D+08 Cs-H matching eliminated = 0.173589D−10 Cp-Vmatching Cpbada = 0.166055D−10 End termination b = 50.000 Ohms End ofresults of calculation step No. = 2 File = 2.CKT *** Files for analysiswith losses 2.CKT topology ATNB RS = 50.00 1 RL = 50.00 10 1 CAP Cas 1 20.32717E−12 2 IND La 2 0 0.20000E−07 0.300E+03 3 CAP Ca 2 0 0.97399E−124 CAP Cst 2 3 0.27906E−12 5 CAP Cpt 3 0 0.28444E−11 6 CAP Cp2 3 50.36442E−11 7 IND Ls2 3 4 0.10450E−06 0.700E+03 8 CAP Cs2 4 50.31160E−12 9 CAP Cpt 5 0 0.52388E−11 10 CAP Cst 5 7 0.40553E−10 11 CAPCp3 7 0 0.36494E−11 12 IND Ls3 7 6 0.10450E−06 0.700E+03 13 CAP Cs3 6 00.31205E−12 14 CAP Cst 7 8 0.38013E−12 15 CAP Cpt 8 0 0.13710E−11 16 CAPCsb 8 9 0.16129E−09 17 IND Lsb 9 10 0.20000E−07 0.300E+03 18 CAP Cpa 100 0.16605E−10

[0348]FIG. 37 shows the calculated responses ignoring losses inresonators and inductances, for prototype and transformed order 8filters with two M-SAW resonators (the responses are then combined), andtaking them into account for the transformed filter. It will be notedthat the insertion loss remains reasonable under these conditions for anarrow filter. As in the previous case, parasite capacitances are takeninto account globally. It will be observed that in the case of agrounded metallic package, for the resonator in a horizontal arm, itwould be possible to subtract these two parasite capacitances from thevertical impedance translation capacitances surrounding the resonator,while in the case of the resonator located in a vertical arm, one ofthem is cancelled and the other has to be subtracted after a T→Πtransformation of one of the two capacitances Ts connected to theresonator.

[0349] We will now mention variant embodiments.

[0350] The proposed embodiment is not frequently used at the moment, andseveral of its advantages are mentioned above. Another advantage is dueto the technology used to make the resonators. It will be noted that thevalues of most capacitances introduced in the transformation are small,and that they can be made directly on the substrate used for making theL-SAW resonators, which further lowers the production cost (lithiumtantalate has fairly high dielectric constants (ε_(r) of the order of40) and are stable as a function of temperature and excessively lowdielectric losses, which gives very high quality capacitances.

[0351] Note also that it could have been decided to make a filter withtwo TEM dielectric resonators to replace the LC circuits. This solutionwould keep many of the advantages of the previous solution for reducingthe influence of undesirable off-band responses of L-SAW resonators andwould lead to lower insertion losses, particularly if half-waveresonators are used. This could be a major advantage when very highperformances are required. The proposed process makes it possible to usetwo resonators with an identical characteristic impedance to reduce theproduction cost.

[0352] Also, as mentioned with respect to the field of application ofthe invention, the same transformation process is applicable to otherknown filter topologies, fairly frequently used to obtain filters withsurface wave resonators.

[0353] Three of these are shown in FIGS. 38a, b and c.

[0354]FIG. 38a shows a configuration comprising resonators only, on thehorizontal branches and on the vertical branches.

[0355]FIG. 38b shows a configuration comprising resonators on thehorizontal branches and capacitances on the vertical branches.

[0356]FIG. 38c shows a configuration comprising resonators on thevertical branches and capacitances on the horizontal branches.

[0357] The first figure (FIG. 38a) that only uses three-elementresonators is very interesting when the objective is to use highperformance responses making good use of the general nature of thetransfer function, and therefore the possibilities offered by the largenumber of resonators involved. Equivalent circuits for resonators in thevarious arms are then fairly different (a little bit like the case inthe example mentioned above). The use of these circuits up to now withpiezoelectric surface or bulk wave resonators is very different fromthese possibilities, since it makes it necessary to use two resonatortypes (one for each arm type), a priori and without transformation. Thislimits the number of infinite attenuation frequencies to two. Theproposed transformation process is a means of assimilating the problemto a small number of resonator types (also one or two in practice) thatcan be made with good performances on known materials but that keep theentire potential of responses possible with this topology (andparticularly the large number of different possible infinite attenuationfrequencies). This must give far better performances and it must alsoenable the correction of imperfections with some types of surface waveresonators, that are difficult to avoid at the moment. In particular, itis possible to make judicious use of attenuation poles to eliminateundesirable responses (off-band “parasite” modes) of some resonators(for example L-SAW on lithium niobate).

[0358] The other two topologies (FIGS. 38b and 38 c) are often used dueto the existence of approximate synthesis methods (approximation of apolynomial response in pass band) in order to use identical or almostidentical resonators. Under these conditions, they lead to asymmetricresponses with a single infinite attenuation frequency (multiple), whichrarely corresponds to a required response and does not really lead tothe best possible use of resonators. The use of the transformationprocess according to the invention has the advantage that it enables theproduction of several infinite attenuation frequencies (at finitefrequency, but all located within the same attenuated band) withidentical resonators.

[0359] 7. We will now describe how the process according to theinvention is applied to the case of connection filters.

[0360] n to p connection filters are composed of filters connecting naccesses (inputs) to p accesses (outputs) and allowing only a finitenumber of frequency bands (usually one only) to pass from an input to anoutput. Fairly frequently, there is only one input access (one to pchannels) or only one output access (n channels to one), the connectionfilter being used to send several signals in different frequency bandsto a transmission medium, or to extract several signals from differentfrequency bands. These filters are usually calculated using numericaloptimisation techniques starting from synthesised filters, to obtain therequired response and a topology favourable for putting one or severalaccesses in common, when they are taken alone. Optimisation is usuallybased on minimising an expression representing the difference from thedesired responses, according to a specific criterion (least squares,least k^(th)). It may also be based on a Remetz type method in the caseof an approximation in the Chebyshev sense. In all cases, it modifiesthe values of components, usually without changing the topology of thedifferent filters, so as to obtain all required connection responses,for example globally characterised by its distribution matrix (Sij). Insome cases, it may be useful to introduce one or more supplementarydipoles at the common point(s).

[0361] We will now give a simple example related to the case of aduplexer (two channels to one or one channel to 2), and will then go onto describe how the process according to the invention is applicable tothe case of connection filters, in the next section.

[0362] Therefore, we will consider a connection filter with two channelsdesigned to separate (or multiplex) signals occupying frequency bandsfrom 500 to 530 MHz and 540 to 570 MHz respectively. The choice of an“equal ripple” response in the pass band (0.2 dB maximum) and channelseparation better than 60 dB, is chosen. The first step is to calculateorder 12 elliptical prototypes that have specifications compatible withthese characteristics. The responses of filters taken in isolation areshown in FIG. 39, and the values of the “prototype” filter elements aregiven in table 20 below. TABLE 20 Zigzag filter order = 6 bp6_515 Zigzagfilter order = 6 bp6_555 Fc = 515 MHz BP 30 MHz Fc = 555 MHz BP 30 MHzEND A RS = 66.61 1 END A RS = 66.74 1 1 CAP Ca 1 0 100.5700p 1 CAP Ca 10 100.4700p 2 IND La 1 0 936.9500p 2 IND La 1 0 808.2000p 3 CAP C1 1 21.7711p 3 CAP C1 1 2 1.6341p 4 IND L2 2 3 5.6261n 4 IND L2 2 3 4.8925n 5CAP C2 2 3 15.0440p 5 CAP C2 2 3 15.0250p 6 CAP C3 3 4 2.0528p 6 CAP C33 4 1.7651p 7 IND L3 4 0 52.5880n 7 IND L3 4 0 52.1940n 8 CAP C4 3 05.6440p 8 CAP C4 3 0 14.5630p 9 CAP C1 3 5 2.3529p 9 CAP C1 3 5 2.1744p10 IND L2 5 6 3.2378n 10 IND L2 5 6 2.806n 11 CAP C2 6 6 26.9510p 11 CAPC2 6 6 26.9260p 12 CAP C3 6 7 813.3700f 12 CAP C3 6 7 699.6100f 13 INDL3 7 0 128.7300n 13 IND L3 7 0 128.0100n 14 CAP C4 6 0 10.4430p 14 CAPC4 6 0 9.7121p 15 CAP Cb 6 8 287.3700f 15 CAP Cb 6 8 247.0400f 16 IND Lb8 9 39.7900n 16 IND Lb 8 9 339.7900n End B RL = 50.00 9 End B RL = 50.009

[0363] The second step is to connect these filters so that they have acommon end B. The characteristics of the responding input (or output)impedance facilitate this connection and there is no need to use acompensation dipole. An optimisation program is used to vary the valuesof the components of the two filters without modifying the topology,such that the response of each of the filters with a common access is asclose as possible, within the pass-band, to the previous response offilters considered in isolation and such that the rejection condition issatisfied. This is done using an optimisation technique based onminimising an error term comprising quadratic errors of transmissions inabout 300 frequencies located in the pass bands and attenuated bands ofthe two filters (least squares approximation of a response that isitself a Chebyshev approximation of an envelope curve. Other techniquesare also known and have been evaluated (direct approximation of aChebyshev envelope curve, least p^(th)s, etc.). The inventors consideredthat the method used in this presentation has the best performances inthis case. FIG. 40 shows the variation of responses in the pass-bandduring the optimisation (all ten optimisation steps starting from the10^(th)) (curves with the greatest variation) up to the 400^(th))). Thereference responses in this figure are shown in dashed lines. FIG. 41shows the responses finally obtained for the duplexer.

[0364] The initial and final values of the elements of the electricalcircuit for the filters are shown in table 21 below, in which it can beseen that most of the modifications are minimal except for elementsclose to the common end. TABLE 21 Values of elements of the circuitafter optimisation: Zigzag filter order = 6 bp6_5152.opt Zigzag filterorder = 6 bp6_5552.opt Fc = 515 MHz BP 30 MHz Fc = 555 MHz BP 30 MHz ENDA RS = 66.61 1 END A′ RS = 66.74 17 1 CAP Ca 1 0 98.7285p 1 CAP Ca 17 098.1422p 2 IND La 1 0 953.4866p 2 IND La 17 0 826.5830p 3 CAP C1 1 21.8070p 3 CAP C1 17 16 1.6505p 4 IND L2 2 3 5.3888n 4 IND L2 16 144.6817n 5 CAP C2 2 3 15.7318p 5 CAP C2 16 14 15.7292p 6 CAP C3 3 42.1062p 6 CAP C3 14 15 1.85662p 7 IND L3 4 0 51.2955n 7 IND L3 15 049.7250n 8 CAP C4 3 0 16.0177p 8 CAP C4 14 0 15.0920p 9 CAP C1 3 52.3769p 9 CAP C1 14 13 2.2258p 10 IND L2 5 6 3.2483n 10 IND L2 13 112.7587n 11 CAP C2 6 6 26.8356p 11 CAP C2 13 11 27.3887p 12 CAP C3 6 7854.4680f 12 CAP C3 11 12 754.2172f 13 IND L3 7 0 122.5328n 13 IND L3 120 118.7264n 14 CAP C4 6 0 11.2187p 14 CAP C4 11 0 10.4311p 15 CAP Cb 6 8287.3700.f 15 CAP Cb 11 10 239.7907f 16 IND Lb 8 9 349.9832n 16 IND Lb10 9 343.5196n End B (common) RL = 50.00 End B (common) RL = 50.00

[0365] Application of the process according to the invention to the caseof connection filters satisfies the criteria given below.

[0366] The steps mentioned above for “simple” filters are applicable toconnection filters composed of filters with minimum inductancetopologies, or using other topologies satisfying criteria definedpreviously. Also in this case, different coil filter technologies can beused, with dielectric resonators and piezoelectric resonators, possiblycombining them in the same connection filter. However, some additionalconsiderations need to be taken into account. In particular, the factthat the two filters have a common end should be taken into account, andif applicable the solution to be selected for impedance transformationsat the common end should be chosen carefully. If narrow band matching ischosen, the choice of the type of matching reactances may be important,but in some cases we may want to choose wide band matching sinceconnection filters with a large number of channels fairly frequentlyoperate on a wide spectrum.

[0367] This will be better understood after reading the followingexample that describes an application of the process to the case of theduplexer considered above. Variants will also be proposed.

[0368] In this example, we want to make the connection filter at a onechannel to two channel connection using a circuit transformation so thatit uses TEM dielectric resonators with the same characteristic impedance(the highest performance solution within this frequency range). Thisdeliberately simplified example is presented here to describe theprinciple according to the process. In practice, we could firstly addadditional refinements (using free parameters) to obtain more practicalvalues for some components as was also mentioned above, and it may alsobe attractive to use another method of obtaining equal terminationresistances at the common access after the circuit transformation (forexample by additional separation of component−inductance L_(b) in casessimilar to this).

[0369] During the calculation mentioned above, an electrical circuit forthe duplexer was produced by optimisation, and the duplexer electricalcircuit and the values of elements were given in table 21. In this case,considering the comments made above, a particular transformation wasmade taking account of the fact that the common end B of filters isterminated on 50 ohms (FIG. 42).

[0370]FIG. 42 shows that if transformation ratios m_(b1) and m_(b2) arechosen to be equal in the steps (for the two filters) of the systematiccircuit transformation process to input lines with the samecharacteristic impedance, particularly for resonators at end B, it willbe possible to eliminate the transformers. This elimination modifies thetermination resistance and as in the case of “simple” filters, it willthen be necessary to match to obtain the required load resistance(source resistance). Therefore, the procedure for these two filters isessentially as described above, but for ends B of the two filters, thecapacitances of the series resonators of the arms joining the commonpoint to the two filters, are separated into only two parts. Thisseparation is made such that the transformation ratios (modifying theimpedance levels of resonators L_(sb1)-C_(sbv1) and L_(sb2)-C_(sbv2) tomake them achievable by a dielectric resonator, for example an opencircuit quarter wave) are equal.

[0371] Thus, in the case of the example considered, the result obtained(having chosen values of parameters that are still free within the rangein which a solution exists) is transformed circuits like thoseillustrated in FIG. 43 in which the two transformation ratios at thecommon end are equal (in this case m_(b1)=m_(b2)=0.120). (FIG. 43 showsa circuit after elimination of the transformers). The responses of theduplexer thus transformed are shown in FIG. 44.

[0372] The final step is a narrow band matching that modifies the levelof the termination resistance at the common point, which gives the finalcircuit illustrated in FIG. 45.

[0373] Tables 22 and 23 below show the values of elements related toFIGS. 43 and 45. TABLE 22 Values of elements of the transformed circuitas far as the common point (end B) Filter Fc = 515 MHz BP 30 MHz FilterFc = 555 MHz BP 30 MHz transformed transformed END A1 R = 50.00 Ohms ENDA2 R = 50.00 Ohms 1 CAP Csa1 1 2 2.8500p 1 CAP Csa2 1 2 2.5294p 2 INDLat1 2 0 4.0861n Res.1_1 2 IND Lat2 2 0 3.8272n 3 CAP Cat1 2 0 19.0590pRes.1_1 3 CAP Cat2 2 0 17.8510p Res.1_2 4 CAP Cst1 2 3 2.2234p 4 CAPCst2 2 3 1.6834p Res.1_2 5 CAP Cpt1 3 0 5.4950p 5 CAP Cpt2 3 0 4.1192p 6CAP Cs21 3 4 19.8830p 6 CAP Cs22 3 4 18.5360p 7 IND Pl21 4 5 2.1314nRes.2_1 7 IND P122 4 5 1.9870n Res.2_2 8 CAP Cp21 4 5 39.7660p Res.2_1 8CAP Cp22 4 5 37.0710p Res.2_2 9 CAP Cpt1 5 0 9.9945p 9 CAP Cpt2 5 010.9440p 10 CAP Cst1 5 6 15.8440p 10 CAP Cst2 5 6 20.6720p 11 CAP Cs31 67 10.0400p 11 CAP Cs32 6 7 9.2798p 12 IND Lp31 7 0 2.1525n Res.3_1 12IND Lp32 7 0 1.9896n Res.3_2 13 IND Cp31 7 0 40.1590p Res.3_1 13 INDCp32 7 0 37.1190p Res.3_2 14 CAP Cst1 6 8 5.3762p 14 CAP Cst2 6 83.5628p 15 CAP Cpt1 8 0 1.3894p 15 CAP Cpt2 8 0 606.8400f 16 CAP Cs41 89 44.8070p 16 CAP Cs42 8 9 41.7170p 17 IND Lp41 9 10 2.1614n Res.4_1 17IND Lp42 9 10 2.0124n Res.4_2 18 CAP Cp41 9 10 40.3260p Res.4_1 18 CAPCp42 9 10 37.5460p Res.4_2 19 CAP Cpt1 10 0 2.2418p 19 CAP Cpt2 10 02.9408p 20 CAP Cst1 10 11 6.9032p 20 CAP Cst2 10 11 12.7430p 21 CAP Cs5111 12 5.9064p 21 CAP Cs52 11 12 5.4619p 22 IND Lp51 12 0 2.2160n 22 INDLp52 12 0 2.0493n Res.5_2 23 CAP Cp51 12 0 41.3450p Res.5_1 23 CAP Cp5212 0 38.2330p Res.5_2 24 CAP Cst1 11 13 3.1591p Res.5_1 24 CAP Cst2 1113 2.5739p 25 CAP Cpt1 13 0 25.1370p 25 CAP Cpt2 13 0 19.8270p 26 CAPCsb1 13 14 62.2100p Res.6_1 26 CAP Csb2 13 14 61.7810p Res.6_2 27 INDLsb1 14 15 5.0750n Res.6_1 27 IND Lsb2 14 15 5.0400n Res.6_2 End BCommon R = 50.00*mb² End B Common R = 50.00*mb² Characteristic impedanceof resonators Characteristic impedance of resonators 11.5 11.5

[0374] TABLE 23 Matching elements [calculation for Faverage =(515*555)^(1/2) MHz and mb² = 0.120] Alternative circuit 1 Lsb_(adapt) =1.779 H Cpb_(adapt) = 49.08 pF Alternative circuit 2 Csb_(adapt) =49.806 pF Lpb_(adapt) = 1.8055 nH

[0375] We will now describe some variants of the process for makingconnection filters.

[0376] The first variant consists of replacing narrow band matching by awider band matching technique, when necessary, using known techniquesthat involve the use of quadripoles that are more complex than thetwo-element quadripoles used above (it is fairly frequent for connectionfilters to operate over a very wide frequency band). Another advantageof wide band matching is that the reactive part of the input impedancecan be taken into account at the common point and thus additionalfiltering can be added outside pass bands of filters forming theconnection. This can be done using high pass, low pass or pass bandstructures.

[0377] A second useful variant in the case of complex connection filtersconsists of carrying out the different operations in a different order.This variant is based partly on the fact that the phase to optimiseconnection filter responses means that it is possible to significantlymodify the termination resistances, without making any important changeto the responses. It is also based on the fact that there are severalsolution infinities (as a result of free parameters) for filtersobtained by transformation, in which all dielectric resonators replacingLC circuits in three-element resonators have the same equivalent circuit(if it was chosen that resonators in horizontal arms will operate withparallel resonance and resonators in vertical arms will operate withserial resonance, their equivalent circuit is fixed by the value of thecharacteristic impedance and conservation of attenuation poles).

[0378] Therefore this variant (for the case of dielectric resonators,but a transposition can be made for cases in which it is required to usea small number of inductances or piezoelectric resonators), consists ofdirectly transforming circuits for filters taken alone by addingmatching elements (without making any attempt to minimise the number ofcomponents by recombination), and then optimising the response of theconnection by leaving the resonators mentioned above fixed and fixinglimits to the values of inductances of end resonators (so that thedielectric resonators can develop a suitable characteristic impedance).In this case, it is possible to not introduce many additional elementsand to have no selective elements between the access and the commonpoint of the different filters, which is essential when the connectioncomprises a high pass filter, pass bands filters and low pass filters.

[0379] Naturally, it is possible to create solutions that use the simpleexample given and the variants indicated above, or even other knownprocesses.

[0380] Application of the process according to the invention forconnection filters is particularly useful in the current context.

[0381] It is observed that with the process according to the invention,it is possible to make connection filters using dielectric resonatorswith the same characteristic impedance and with the most generalresponses possible using the different zigzag filter topologies.

[0382] More generally, considering the current interest in findingeconomic, compact (miniaturisation) and high performance solutions tothe question of antenna filtering in radio communication terminals thatare becoming multistandard and multi-band [for example GSM 900 MHz+GSM1800 MHz+IMT2000 2 GHz+IMT2000 2.5 GHz+Satellite], this process isbecoming particularly useful for making more general connection filtersenabling the connection of few and possibly multi-band antennasrespectively to reception and transmission stages (see example in FIG.46).

[0383]FIG. 46 diagrammatically shows three antennas. The first antennais connected using a “band 1r” connection filter at the input to aconverter/LNA amplifier which is connected to a receiving stage througha network comprising three filters in parallel, called the “band 1r”,“band 2r” and “band 3r” filters. The second antenna is connected firstlyto the LNA input through a “band 2r” connection filter, and secondly tothe output of a converter/PA amplifier through a “band 2e” connectionfilter. The input of the element PA is connected to a transmitter stagethrough a network comprising two “band 2e” and “band 3e” connectionfilters in parallel. The third antenna is connected firstly to the LNAinput through a “band 3r” connection filter, and secondly to the PAoutput through a “band 3e” connection filter.

[0384] This process can also be used to make connection filters usingbulk or surface wave piezoelectric resonators and/or dielectricresonators. It appears that maximum use of the first of thesetechnologies (piezoelectric resonators) is the most promising way ofachieving performance and size gains due to the “natural” size of theseresonators and also due to possibilities of integrating complexfunctions on the same substrate (several filters forming all or part ofa connection filter).

[0385] The process can also be used to make filters with severalseparate or contiguous pass bands (filter connections through bothends), for example by using a single type of surface wave resonators(per pass band) that can lead to compact, inexpensive, integratablesolutions for inter-stage filtering in multi-band terminals. The specialcase of contiguous or overlapping pass bands can use surface wavetechnologies or thin piezoelectric layer technologies and be one of themeans for making compact and low cost filters capable of carryinggreater power while retaining high performances. This means may becombined with previously known solutions (use of highly parallelstructures) to further increase the allowable power in these filters.

[0386]FIG. 47 shows another example indicating functions that could beimplemented using these concepts in a mobile terminal or in amulti-frequency and multi-standard base station.

[0387] In this example, simple structures were chosen for connectionfilters that would be compatible with monolithic integration of theentire radio frequency (UHF) part of a mobile terminal or a base stationon a semiconductor, if filters involving thin piezoelectric layers areused.

[0388]FIG. 47 diagrammatically shows a dual band antenna connected to areceiver stage by means of a circuit comprising two branches inparallel, one comprising a “band lr” connection filter, aconverter/amplifier LNA1 and a connection filter forming a “band 1r”duplexer in series, the other comprising a “band 2r” connection filter,a converter/amplifier LNA2 and a connection filter forming a “band 2r”duplexer in series. Furthermore, the antenna is connected to atransmitter stage through a circuit comprising two branches in parallel,one comprising a connection filter with a “band 1e” duplexer, aconverter/amplifier PA1 and a connection filter forming a “band 1e”multiplexer in series, the other comprising a “band 2e” connectionfilter with a converter/amplifier PA2 and a connection filter forming a“band 2e” duplexer in series. The various means thus described form anintegratable UHF subassembly.

[0389] In particular, the process according to this invention can beused on bulk or surface wave crystal resonator filters thinpiezoelectric layer resonator filters, or piezoelectric resonatorfilters using non-crystalline materials such that the ratios of Cparallel/C series elements of the equivalent resonators circuit remainwithin a given range and are set equal to a given value, and that theinductances of these resonators are equal to a given value adapted toeconomic, easy and high performance production of these resonators.

[0390] This invention may also be used for the design of matchablefilters for which it is desired to limit the variation of the couplingcapacitances between resonators while keeping a pass band with anapproximately constant value (in MHz and not as a relative value in %).

[0391] It will be observed that the topology of transformed filtersconform with this invention is characterised particularly by thesystematic presence of capacitance T or Π between resonators.

[0392] It should also be noted that after the transformation conformwith this invention described above, the topology of the circuitobtained can be described by additional simple transformations, whichcan have a real advantage in some cases of inductance and capacitancefilters. For example, a capacitances Π that appears between resonantcircuits can be transformed into a capacitances T, and a capacitances Tthat appears between resonant circuits can be transformed into acapacitances Π. The advantage of this transformation is that it modifiesthe values of capacitances in the right direction.

[0393]FIG. 48 diagrammatically shows this transformation of T or Πsub-circuits into Π or T sub-circuits, respectively.

[0394]FIG. 48a shows the capacitance of a first vertical branch denotedCa, the capacitance of a horizontal branch denoted Cb and thecapacitance of a second vertical branch of a Π denoted Cc. In FIG. 48b,the two capacitances of the horizontal branch are denoted C1 and C2, andthe capacitance of the vertical branch of a T is denoted C3.

[0395] The transformation is obtained as follows:

C ₁=(C _(a) C _(b) +C _(b) C _(c) +C _(a) C _(c))/C _(a)

C ₂=(C _(a) C _(b) +C _(b) C _(c) +C _(a) C _(c))/C _(b)

C ₃=(C _(a) C _(b) +C _(b) C _(c) +C _(a) C _(c))/C _(c) and

C _(a)=C₁ C ₂/(C ₁ +C ₂ +C ₃)

C _(b) =C ₁ C ₃/(C ₁ +C ₂ +C ₃)

C _(c) =C ₂ C ₃/(C ₁ +C ₂ +C ₃)

[0396] The transformation process described above according to thisinvention is most useful in the case in which the dipoles forming thearms of the prototype ladder filter are resonators that comprise notmore than one inductance (the value of this inductance can betransformed to obtain a required value). Thus preferably, this inventionis applicable to prototype filters for which the arms only comprise oneelement (one capacitance except at the ends) or only comprise aresonator with two or three elements and a single inductance. Obviously,it is also possible to make prior transformations to reach these cases.It is also possible to obtain prototype circuits comprising irreducibledipoles (multimode resonators) with 3, 4 or more elements including morethan one inductance (for example for prototypes obtained by lowpass-pass band transformations). However, it is generally possible tomake transformations considering several consecutive arms of the filter,to create an equivalent structure comprising only arms composed of acapacitance or resonators with a single inductance (Saal and Ulbrichtransformation, Colin transformation, etc.).

[0397] Furthermore, the same type of transformation as that describedabove is possible by dividing inductances. The use of two types oftransformation then enables useful generalisations.

[0398] In the above description, the capacitances are separated intothree parts to eliminate internal transformers (or possibly into twoparts in some cases for the ends) by a Norton transformation (like thatshown diagrammatically particularly in FIGS. 7a and 7 b). However, thisinvention is not limited to these particular transformations. FIGS. 49and 50 diagrammatically show equivalent transformations that could beused within the context of this invention. If these transformationsresult in negative capacitances, they should be recombined with similarpositive capacitances.

[0399]FIG. 49 diagrammatically shows the transformation of a circuitcomprising an impedance Z in parallel on the input of a transformer withratio 1/n into a T circuit comprising a first impedance (1−n)Z and asecond impedance n(n−1)Z in its horizontal branch, and an impedance nZin its vertical branch.

[0400]FIG. 50 diagrammatically shows the transformation of a circuitcomprising an impedance Z in series on the input to a transformer with aratio n/1 into a Π circuit comprising an impedance Z/(1−n) in its firstvertical branch, an impedance Z/n in its horizontal branch, and animpedance Z/n(n−1) in its second vertical branch.

[0401] Obviously, this invention is not limited to the particularembodiments that have just been described, but can be extended toinclude any variants conform with its spirit.

[0402] In particular, this invention is applicable to the design ofcomponents for mobile radio communications and satellite communicationswithin the 0.5-3 GHz frequency range (filters and multiplexers usingsurface wave resonators or dielectric resonators in particular) andtransmission by cable. It also enables new solutions for filtering inmulti-band or multi-standard terminals.

BIBLIOGRAPHY

[0403] Filter Classification, Properties and Synthesis

[0404] [1] J. K Skwirzynski. On the synthesis of filters IEEE Trans.Circuit Theory vol. CT-18 p 152-16 Jan, 3, 1971.

[0405] [2] R. Saal, E. Ulbrich. On the design of filters by synthesis.IRE Trans. on circuit theory CT-5 pp 284-317, December 1958

[0406] [3] A. I. Zverev. Handbook of filter synthesis. Wiley New York1967

[0407] [4] P. Amstutz. Elliptic approximation and elliptic filter designon small computers. IEEE Trans. on Circuits and Systems vol. CAS25 pp1001-1011 1978

[0408] [5] M. Hasler, J. Neyrinck. Filtres Electriques (ElectricalFilters). Electricity treatise volume XIX. Presses UniversitairesRomandes, Lausanne 1982

[0409] [6] P. Bildstein, Synthèse des filtres LC (Synthesis of LCfilters). Doc E3120-3, Engineering Techniques. Electronics Treatise volE3, 21p.

[0410] Filters Using Impedance Inverters

[0411] [7] P. Amstutz. Filtres à bandes étroites (Narrow band filters).Cable and Transmission; vol. 2 pp 88-97, 1967

[0412] [8] R. J Cameron. Fast generation of Chebyshev filter prototypeswith asymmetrically-prescribed transmission zeros. ESA Journal vol. 6,pp 83-95, 1982

[0413] [9] R. J Cameron. General prototype Network Synthesis method formicrowave filters. ESA Journal vol.6, pp 196-206, 1982

[0414] [10] G. Macchiarella. An original approach to the design ofbandpass cavity filters with multiple couplings. IEEE Trans. MTT, vol.45No. 2, pp 179-187, February 1997.

[0415] Dielectric Resonators, Particularly TEM or Quasi TEM Resonators,and Filters Using Them for Applications to Cell Phones:

[0416] [11] D; Kahfez, P. Guillon. Dielectric resonators. Artech House,London 1986

[0417] [12] C. Veyres: Circuits pour Hyperfréquence (Circuits forHyperfrequency) (Documents E620.3, E621.1, E622.1, E623.1, E624.1,E625.1; Techniques de l'ingénieur, Traité d'électronique (EngineeringTechniques, Electronics Treatise): Vol E1.

[0418] [13] Jeong-Soo Lim, Dong Chui Park. A modified Chebyshev BandpassFilter with attenuation poles in the Stopband. IEEE Trans. MTT, vol. 45No. 6, pp 898-904, June 1997.

[0419] [14] T. Nishikawa. RF front end circuit components—TelephonesIEICE Transactions E 74 No. 6, pp 156-162, June 1991

[0420] [15] H. Matsumoto et al. Miniaturized duplexer. Telephone IEICETransactions E 74 No. 5, pp 1214-1220, May 1991

[0421] Principle and Properties of Crystal Resonators and Filters UsingCrystal Resonators with Bulk or Surface Waves:

[0422] [16] D; Royer, E. Dieulesaint. Ondes Elastiques dans les Solides(Elastic waves in Solids), volumes 1 & 2, Masson, Paris, 1999.

[0423] [17] E. A Gerber, A. Ballato, Precision frequency Control, volume1, chapter 5, Piezoelectric and Electromechanical filters. AcademicPress, New York 1985

[0424] [18] R. G. Kinsman. Crystal Filters John Wiley, New York 1987

[0425] [19] J. Détaint, Optimisation of AT berlinite and quartzthickness shear resonators for VHF filter applications. Proc. 1991 IEEEInt. Frequency Control Symposium, pp 166-180

[0426] [20] M. Feldmann, J. Henaff. Dispositif de Traitement du Signal àOndes de Surface (Surface Wave Signal Processing Device), Masson, Paris1983

[0427] [21] C. C. W. Ruppel, R. Dill, A. Fisherauer, G. Fisherauer etal. SAW devices for consumer communication applications. IEEE Trans.Ultrason. Ferroelectrics & Frequency Control V. 40 No. 5, pp 438-450,1993

[0428] [22] C. K. Campbell. Surface Acoustic Wave for Mobile andWireless Communications. Academics Press, Boston, 1998.

[0429] Circuit Theory and Circuit Transformations

[0430] [23] M. Feldmann. Théorie des réseaux et des systèmes linéaires(Theory of networks and linear systems). Eyrolles, Paris, 1981)

[0431] [24] A. B. Williams, F. J. Taylor. Electronic Filter DesignHandbook. MacGraw Hill, N.Y., 1992

[0432] [25] S. Stephanescu. Filtres Electriques (Electric filters).Masson. Paris, 1972

[0433] [26] J. E. colin. Transformation des quadripoles permettantl'introduction des cristaux piezo-électriques dans les filtrespasse-bande en échelle (Transformation of quadripoles enabling theintroduction of piezoelectric crystals in ladder pass band filters).Cable et Transmission (Cable and Transmission) v.21 No. 2 (1967) (Manyother articles on circuit transformations by this author).

[0434] Connection Filters

[0435] [27] G. Szentimaï (editor). Computer aided filter design. IEEEpress 1973. (Collection of basic articles on different questions aboutthis theme including several by J. W Bandler).

[0436] [28] J. W. Bandler, S. Daijavad, Q. J. Zhang, “Exact simulationand sensitivity analysis of multiplexing network”. IEEE Trans. MicrowaveTheory, vol. MMT-34, pp 93-102, 1986

[0437] [29] M. K Chahine. Conception de multiplexeur hyperfréquence pourcharge utile de satellite de télécommunication (Design of hyperfrequencymultiplexer for telecommunication satellite payload). University ofParis VI thesis presented on May 12, 1994.

1. Process for optimisation of the elements of a narrow or intermediatepass band filter for which the LC prototype has been determined,characterised by the fact that it includes the following steps: i)decompose resonators with X elements into several parallel or seriescapacitances ii) insert pairs of transformers between the first separateelement and the rest of the resonator, iii) displace transformers tomodify resonator impedance levels, and iv) absorb residual transformersby transformation.
 2. Process according to claim 1, characterised by thefact that the decomposition step i) consists of decomposing X-elementresonators into two parallel or series capacitances.
 3. Processaccording to claim 1, characterised by the fact that the decompositionstep i) consists of decomposing X-element resonators into three parallelor series capacitances.
 4. Process according to one of claims 1 to 3,characterised by the fact that the decomposition step i) is applicableto two-element resonators.
 5. Process according to claim 4,characterised by the fact that the decomposition step i) consists ofreplacing a parallel circuit comprising an inductance L_(p) in parallelwith a capacitance C_(p) by a circuit comprising an inductance L_(p) inparallel with three capacitances C_(pu), C_(pv) and C_(pw).
 6. Processaccording to claim 4, characterised by the fact that the decompositionstep i) consists of replacing a series circuit comprising an inductanceLs in series with a capacitance C_(s) by a circuit comprising aninductance Ls in series with three capacitances C_(su), C_(sv) andC_(sw).
 7. Process according to one of claims 1 to 3, characterised bythe fact that the decomposition step i) is applicable to three-elementresonators.
 8. Process according to claim 7, characterised by the factthat the decomposition step i) consists of replacing a circuitcomprising a capacitance C_(s) in series with a resonator formed from aninductance M_(p) in parallel with a capacitance C_(p), by a circuitcomprising two capacitances C_(su), C_(sw), in series on the input sideof a resonator formed from an inductance M_(p) in parallel with acapacitance C_(p) and a third capacitance C_(sw) in series on the outputside of this resonator.
 9. Process according to claim 7, characterisedby the fact that the decomposition step i) consists of replacing acircuit comprising a capacitance Cp in parallel with a resonator formedby an inductance Ls in series with a capacitance Cs, by a circuitcomprising three capacitances Cpu, Cpv and Cpw in parallel with aresonator formed by an inductance Ls in series with a capacitance Cs.10. Process according to one of claims 1 to 9, characterised by the factthat the transformer insertion step ii) consists of inserting pairs oftransformers with ratios 1/ml and mi/1.
 11. Process according to one ofclaims 1 to 10, characterised by the fact that the transformerdisplacement step iii) consists of making transformers with ratios suchas m₁/m₂=1, then m₂/m₃=1 then m₃/m₄ =1, . . . , etc. appear.
 12. Processaccording to one of claims 1 to 11, characterised by the fact that thetransformer elimination step iv) consists of replacing a transformer andtwo capacitances, by two other capacitances.
 13. Process according toclaim 12, characterised by the fact that the transformer eliminationstep iv) consists of replacing a circuit comprising a parallelcapacitance C_(p) on the input side and a parallel transformer on theoutput side, connected through a series capacitance C_(s) on the headline by a circuit comprising a series capacitance C_(st) on the inputside and a parallel capacitance C_(pt) on the output side.
 14. Processaccording to claim 12, characterised by the fact that the transformerelimination step iv) consists of replacing a circuit comprising a seriescapacitance C_(p) on the input side and a parallel capacitance C′p onthe input side of a parallel transformer on the output side, by acircuit comprising a parallel capacitance C′_(pt) on the input side anda series capacitance C′_(st) on the output side.
 15. Process accordingto one of claims 1 to 14, characterised by the fact that the transformerelimination step iv) also consists of eliminating end transformers. 16.Process according to claim 15, characterised by the fact that the endtransformer elimination step iv) consists of replacing an endtransformer by two impedances with opposite signs.
 17. Process accordingto either claim 15 or 16, characterised by the fact that during the stepto eliminate an end transformer, one of the end components is absorbedby this transformation or is replaced by another component with the samenature and with a different value.
 18. Process according to one ofclaims 15 to 17, characterised by the fact that the end transformerelimination step iv) consists of doing a match using two impedances Z₁and Z₂ with opposite signs, defined by: $\begin{matrix}{Z_{1} = {{\pm j}\sqrt{Z_{0}\left( {Z^{*} - Z_{0}} \right)}}} & \quad & {and} & \quad & {Z_{2} = {{\pm j}\quad Z^{*}{\sqrt{\frac{Z_{0}}{Z^{*} - Z_{0}}}.}}}\end{matrix}$

and in these relations: Z₀ is the required termination impedance, and Z*is the termination impedance after displacement of transformers at theends.
 19. Process according to one of claims 15 to 17, characterised bythe fact that the end transformer elimination step iv) consists ofmaking a match using two impedances Z₁ and Z₂ with opposite signs,defined by: $\begin{matrix}{Z_{1} = {{\pm j}\sqrt{Z_{0}\left( {Z^{*} - Z_{0}} \right)}}} & \quad & {and} & \quad & {Z_{2} = {{\mp j}\quad Z_{0}{\sqrt{\frac{Z^{*}}{Z_{0} - Z^{*}}}.}}}\end{matrix}$

relations in which: Z₀ is the required termination impedance, and Z* isthe termination impedance after displacement of transformers at theends.
 20. Process according to one of claims 18 or 19, characterised bythe fact that one of the impedances (Z1) is a capacitance, while theother impedance (Z2) is a negative capacitance.
 21. Process according toone of claims 18 or 19, characterised by the fact that one of theimpedances (Z1) is a capacitance, while the other impedance (Z1) is aparallel inductance.
 22. Process according to any one of claims 1 to 21,characterised by the fact that it uses a two-step optimisation process:a first step that consists of choosing some auxiliary parameters such asthe ratios of the capacitances of three-element resonators and one ofthe capacitances output from the decomposition of the two-elementresonator at one end, to determine the range in which a solution existsfor at least one chosen value of the inductances, and a second stepwhich consists of limiting variations of parameters in this range,starting from one end of the filter, to solve equations by choosing atleast one of the impedance translation capacitances located between theresonators from a table of standard values, using an algorithm makinguse of the calculated value at the centre of the range.
 23. Processaccording to one of claims 1 to 22, characterised by the fact that forthe design of a multiplexer with several filters and for the ends of thefilters, the capacitances of the series resonators of the arms joiningthe point common to the two filters is divided into two parts, and thisdivision is made such that the transformation ratios are equal. 24.Process according to one of claims 1 to 23, characterised by the factthat it is used for the design of filters based on bulk wave resonators.25. Process according to one of claims 1 to 23, characterised by thefact that it is used for the design of filters based on surface waveresonators.
 26. Process according to one of claims 1 to 25,characterised by the fact that it is used for the design of filtersbased on piezoelectric crystal resonators, for example with lithiumtantalate, or thin piezoelectric layers, for example zinc oxide. 27.Process according to one of claims 1 to 26, characterised by the factthat it is used for the design of a pass band filter for which all theinductances are made equal to a fixed value.
 28. Process according toany one of claims 1 to 27, characterised by the fact that it is used forthe design of a pass band filter with dielectric resonators in TEM mode.29. Process according to one of claims 1 to 28, characterised by thefact that it is used with at least one resonator in series, one of theaccesses being on the central core of the resonator and the other accessof the quadripole being on the external metallisation of the resonator.30. Process according to one of claims 1 to 29, characterised by thefact that it is used for the design of a filter in which at least somecapacitances are brought to fixed values, for example standardized. 31.Process according to one of claims 1 to 30, characterised by the factthat it is used for the design of a filter with dielectric resonatorswith the same characteristic impedance.
 32. Process according to one ofclaims 1 to 31, characterised by the fact that it is used for the designof a filter with piezoelectric resonators with at least approximatelythe same ratio of capacitances and at least similar inductances. 33.Process according to one of claims 1 to 32, characterised by the factthat it is used for the design of a hybrid filter or multiplexer usinginductances and dielectric resonators and/or piezoelectric resonators.34. Process according to one of claims 1 to 33, characterised by thefact that it is used for the design of a ladder pass band filter. 35.Process according to one of claims 1 to 34; characterised by the factthat it is used for the design of a polynomial filter formed of asuccession of series LC dipoles in the horizontal arms and parallel LCdipoles in the vertical arms.
 36. Process according to one of claims 1to 35, characterised by the fact that it is used on bulk or surface wavepiezoelectric crystal resonator filters, or piezoelectric resonatorsusing non-crystalline materials, such that the ratios of parallelC/series C elements in the equivalent resonator circuit remain within agiven range or are equal to a given value, and that the inductances ofthese resonators are equal to a given value adapted to economic, easyand high performance production of these resonators.
 37. Processaccording to any one of claims 1 to 36, characterised by the fact thatit is used to make connection filters composed of filters connecting ninput accesses to p output accesses, only allowing a finite number offrequency bands to pass from one input to one output.
 38. Processaccording to one of claims 1 to 37, characterised by the fact that it isused for the design of matchable filters for which it is required tolimit the variation of coupling capacitances between resonators whilekeeping a pass band with an approximately constant value.
 39. Processaccording to one of claims 1 to 38, characterised by the fact that theoptimisation applies to a circuit composed of only part of a filter. 40.Process according to one of claims 1 to 37, characterised by the factthat it comprises the step consisting of transforming a T circuit into aΠ circuit.
 41. Process according to one of claims 1 to 37, characterisedby the fact that it comprises the step consisting of transforming a Πcircuit into a T circuit.
 42. Pass band filter obtained by using theprocess conform with one of claims 1 to
 41. 43. Filters conform withclaim 42, characterised by the fact that they are grouped for makingpass band multiplexers formed from several filters connecting n inputaccesses to p outputs, only allowing a finite number of frequency bandsto pass from an input to an output.
 44. Filters according to claim 43,characterised by the fact that they comprise a 2-channel to 1-channelduplexer or a 1-channel to 2-channel duplexer.